Number of problems in this table is 1968
Column notations: A is ODE degree. B is Program Number of solutions generated. C is CAS Number of solutions generated.
|
# |
ODE |
A |
B |
C |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = x^{2}-1 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.924 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (2+x \right ) y^{\prime }+\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.341 |
|
|
\[ {}\left (1+x \right ) y^{\prime \prime }-\left (2+x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.344 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[_Gegenbauer] |
✓ |
✓ |
0.363 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[_Gegenbauer] |
✓ |
✓ |
0.349 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.385 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-6 x y^{\prime }+12 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer] |
✓ |
✓ |
0.503 |
|
|
\[ {}\left (x^{2}+3\right ) y^{\prime \prime }-7 x y^{\prime }+16 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.39 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+8 x y^{\prime }+12 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer] |
✓ |
✓ |
0.588 |
|
|
\[ {}3 y^{\prime \prime }+x y^{\prime }-4 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.676 |
|
|
\[ {}5 y^{\prime \prime }-2 x y^{\prime }+10 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.508 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }-3 y x = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.461 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.359 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }-2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.127 |
|
|
\[ {}\left (x^{2}-6 x +10\right ) y^{\prime \prime }-4 \left (x -3\right ) y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.245 |
|
|
\[ {}\left (x^{2}+6 x \right ) y^{\prime \prime }+\left (3 x +9\right ) y^{\prime }-3 y = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.72 |
|
|
\[ {}t y^{\prime \prime }+\left (t^{2}-1\right ) y^{\prime }+t^{3} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.023 |
|
|
\[ {}t^{2} y^{\prime \prime }-t \left (2+t \right ) y^{\prime }+\left (2+t \right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.694 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.306 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (x -\frac {3}{16}\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.24 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.309 |
|
|
\[ {}t^{2} y^{\prime \prime }-t \left (2+t \right ) y^{\prime }+\left (2+t \right ) y = 2 t^{3} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.737 |
|
|
\[ {}t y^{\prime \prime }-\left (1+t \right ) y^{\prime }+y = t^{2} {\mathrm e}^{2 t} \] |
1 |
1 |
1 |
kovacic, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.607 |
|
|
\[ {}\left (1-t \right ) y^{\prime \prime }+t y^{\prime }-y = 2 \left (t -1\right )^{2} {\mathrm e}^{-t} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.885 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = g \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.22 |
|
|
\[ {}t y^{\prime \prime }-\left (1+t \right ) y^{\prime }+y = t^{2} {\mathrm e}^{2 t} \] |
1 |
1 |
1 |
kovacic, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.512 |
|
|
\[ {}\left (1-t \right ) y^{\prime \prime }+t y^{\prime }-y = 2 \left (t -1\right ) {\mathrm e}^{-t} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.841 |
|
|
\[ {}\left (x^{2}+2\right ) y^{\prime \prime }-x y^{\prime }+4 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.486 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.782 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.282 |
|
|
\[ {}\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.445 |
|
|
\[ {}2 y^{\prime \prime }+x y^{\prime }+3 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.447 |
|
|
\[ {}\left (x^{2}+2\right ) y^{\prime \prime }-x y^{\prime }+4 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.042 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.974 |
|
|
\[ {}\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.089 |
|
|
\[ {}\left (x^{2}+2\right ) y^{\prime \prime }-x y^{\prime }+4 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.04 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.977 |
|
|
\[ {}\left (-x^{2}+4\right ) y^{\prime \prime }+x y^{\prime }+2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.341 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (-16 x^{2}+3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.528 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.941 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.979 |
|
|
\[ {}4 x^{2} \sin \left (x \right ) y^{\prime \prime }-4 x \left (x \cos \left (x \right )+\sin \left (x \right )\right ) y^{\prime }+\left (2 x \cos \left (x \right )+3 \sin \left (x \right )\right ) y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.745 |
|
|
\[ {}\left (x^{2}-2 x \right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }+\left (2 x -2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.866 |
|
|
\[ {}\left (1+2 x \right ) y^{\prime \prime }-2 y^{\prime }-\left (2 x +3\right ) y = \left (1+2 x \right )^{2} \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.247 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+\left (-8 x^{2}+4 x \right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y = 4 \sqrt {x}\, {\mathrm e}^{x} \left (1+4 x \right ) \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.715 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 8 \,{\mathrm e}^{-x \left (2+x \right )} \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.26 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x \left (-1+x \right ) y^{\prime }+\left (x^{2}-2 x +2\right ) y = x^{3} {\mathrm e}^{2 x} \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.256 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (2 x -1\right ) y^{\prime }+\left (x^{2}-x -1\right ) y = x^{2} {\mathrm e}^{x} \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.224 |
|
|
\[ {}\left (-2 x +1\right ) y^{\prime \prime }+2 y^{\prime }+\left (2 x -3\right ) y = \left (4 x^{2}-4 x +1\right ) {\mathrm e}^{x} \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.242 |
|
|
\[ {}2 x y^{\prime \prime }+\left (1+4 x \right ) y^{\prime }+\left (1+2 x \right ) y = 3 \sqrt {x}\, {\mathrm e}^{-x} \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.251 |
|
|
\[ {}x y^{\prime \prime }-\left (1+2 x \right ) y^{\prime }+\left (1+x \right ) y = -{\mathrm e}^{-x} \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.24 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x \left (1+x \right ) y^{\prime }+\left (2 x +3\right ) y = 4 x^{\frac {5}{2}} {\mathrm e}^{2 x} \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.254 |
|
|
\[ {}x y^{\prime \prime }+\left (2-2 x \right ) y^{\prime }+\left (x -2\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.36 |
|
|
\[ {}x^{2} \ln \left (x \right )^{2} y^{\prime \prime }-2 x \ln \left (x \right ) y^{\prime }+\left (2+\ln \left (x \right )\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.139 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +2\right ) y^{\prime }+\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.296 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.333 |
|
|
\[ {}x y^{\prime \prime }-\left (1+4 x \right ) y^{\prime }+\left (4 x +2\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.352 |
|
|
\[ {}4 x^{2} \sin \left (x \right ) y^{\prime \prime }-4 x \left (x \cos \left (x \right )+\sin \left (x \right )\right ) y^{\prime }+\left (2 x \cos \left (x \right )+3 \sin \left (x \right )\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.557 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (-16 x^{2}+3\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.351 |
|
|
\[ {}\left (1+2 x \right ) x y^{\prime \prime }-2 \left (2 x^{2}-1\right ) y^{\prime }-4 \left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.789 |
|
|
\[ {}\left (x^{2}-2 x \right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }+\left (2 x -2\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.421 |
|
|
\[ {}x y^{\prime \prime }-\left (1+4 x \right ) y^{\prime }+\left (4 x +2\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.374 |
|
|
\[ {}\left (3 x -1\right ) y^{\prime \prime }-\left (2+3 x \right ) y^{\prime }-\left (6 x -8\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.513 |
|
|
\[ {}\left (1+x \right )^{2} y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }-\left (x^{2}+2 x -1\right ) y = \left (1+x \right )^{3} {\mathrm e}^{x} \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.424 |
|
|
\[ {}\left (3 x -1\right ) \left (y^{2}+y^{\prime }\right )-\left (2+3 x \right ) y-6 x +8 = 0 \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_rational, _Riccati] |
✓ |
✓ |
2.23 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+\left (-8 x^{2}+4 x \right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y = 4 \sqrt {x}\, {\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.615 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 4 \,{\mathrm e}^{-x \left (2+x \right )} \] |
1 |
1 |
1 |
kovacic, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.281 |
|
|
\[ {}\left (1+2 x \right ) y^{\prime \prime }-2 y^{\prime }-\left (2 x +3\right ) y = \left (1+2 x \right )^{2} {\mathrm e}^{-x} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.668 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +2\right ) y^{\prime }+\left (2+x \right ) y = 6 x^{3} {\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.362 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = x^{3} \cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.809 |
|
|
\[ {}\sin \left (x \right ) y^{\prime \prime }+\left (2 \sin \left (x \right )-\cos \left (x \right )\right ) y^{\prime }+\left (\sin \left (x \right )-\cos \left (x \right )\right ) y = {\mathrm e}^{-x} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
1.198 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (-16 x^{2}+3\right ) y = 8 x^{\frac {5}{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.68 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}+3\right ) y = x^{\frac {7}{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.696 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }-\left (x^{2}-2\right ) y = 3 x^{4} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.66 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x \left (1+x \right ) y^{\prime }+\left (x^{2}+2 x +2\right ) y = x^{3} {\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.396 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x \left (2+x \right ) y^{\prime }+\left (x^{2}+4 x +6\right ) y = 2 x \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.872 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (x^{2}+6\right ) y = x^{4} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.665 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 2 \left (-1+x \right )^{2} {\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.878 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x \left (1+x \right ) y^{\prime }+\left (2 x +3\right ) y = x^{\frac {5}{2}} {\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.786 |
|
|
\[ {}\left (3 x -1\right ) y^{\prime \prime }-\left (2+3 x \right ) y^{\prime }-\left (6 x -8\right ) y = \left (3 x -1\right )^{2} {\mathrm e}^{2 x} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.839 |
|
|
\[ {}\left (2+x \right ) y^{\prime \prime }+x y^{\prime }+3 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.948 |
|
|
\[ {}x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (x +4\right ) y^{\prime }+\left (2-x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.408 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (1+2 x \right ) y^{\prime }-\left (4+6 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.706 |
|
|
\[ {}x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+x \left (2 x^{2}+4\right ) y^{\prime }+2 \left (-x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.944 |
|
|
\[ {}x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+2 x \left (x^{2}+5\right ) y^{\prime }+2 \left (-x^{2}+3\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.483 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+6 x y^{\prime }+6 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.408 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.316 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-8 x y^{\prime }+20 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.323 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-8 x y^{\prime }-12 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer] |
✓ |
✓ |
0.707 |
|
|
\[ {}\left (2 x^{2}+1\right ) y^{\prime \prime }+7 x y^{\prime }+2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.509 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-5 x y^{\prime }-4 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer] |
✓ |
✓ |
1.109 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-10 x y^{\prime }+28 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.344 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.371 |
|
|
\[ {}\left (2 x^{2}+1\right ) y^{\prime \prime }-9 x y^{\prime }-6 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.335 |
|
|
\[ {}\left (2 x^{2}-8 x +11\right ) y^{\prime \prime }-16 \left (x -2\right ) y^{\prime }+36 y = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.842 |
|
|
\[ {}y^{\prime \prime }+\left (x -3\right ) y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.157 |
|
|
\[ {}\left (x^{2}-8 x +14\right ) y^{\prime \prime }-8 \left (x -4\right ) y^{\prime }+20 y = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.815 |
|
|
\[ {}\left (2 x^{2}+4 x +5\right ) y^{\prime \prime }-20 \left (1+x \right ) y^{\prime }+60 y = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.851 |
|
|
\[ {}\left (x^{3}+1\right ) y^{\prime \prime }+7 x^{2} y^{\prime }+9 y x = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.255 |
|
|
\[ {}\left (2 x^{5}+1\right ) y^{\prime \prime }+14 x^{4} y^{\prime }+10 x^{3} y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
28.735 |
|
|
\[ {}y^{\prime \prime }+x^{6} y^{\prime }+7 x^{5} y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.538 |
|
|
\[ {}\left (x^{8}+1\right ) y^{\prime \prime }-16 x^{7} y^{\prime }+72 x^{6} y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.153 |
|
|
\[ {}y^{\prime \prime }+x^{5} y^{\prime }+6 x^{4} y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.54 |
|
|
\[ {}\left (1+3 x \right ) y^{\prime \prime }+x y^{\prime }+2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.61 |
|
|
\[ {}\left (3 x^{2}+x +1\right ) y^{\prime \prime }+\left (2+15 x \right ) y^{\prime }+12 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.048 |
|
|
\[ {}\left (2+x \right ) y^{\prime \prime }+\left (1+x \right ) y^{\prime }+3 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.215 |
|
|
\[ {}\left (x +4\right ) y^{\prime \prime }+\left (2+x \right ) y^{\prime }+2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.245 |
|
|
\[ {}\left (2 x^{2}+3 x \right ) y^{\prime \prime }+10 \left (1+x \right ) y^{\prime }+8 y = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.495 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (6-7 x \right ) y^{\prime }+8 y = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.213 |
|
|
\[ {}\left (2 x^{2}+x +1\right ) y^{\prime \prime }+\left (1+7 x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.836 |
|
|
\[ {}\left (x +3\right ) y^{\prime \prime }+\left (1+2 x \right ) y^{\prime }-\left (2-x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.789 |
|
|
\[ {}y^{\prime \prime }+3 x y^{\prime }+\left (2 x^{2}+4\right ) y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.177 |
|
|
\[ {}\left (4 x +2\right ) y^{\prime \prime }-4 y^{\prime }-\left (4 x +6\right ) y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.306 |
|
|
\[ {}y^{\prime \prime }-3 x y^{\prime }+\left (2 x^{2}+5\right ) y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.293 |
|
|
\[ {}2 y^{\prime \prime }+5 x y^{\prime }+\left (2 x^{2}+4\right ) y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.131 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.074 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.004 |
|
|
\[ {}2 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+x \left (11 x^{2}+11 x +9\right ) y^{\prime }+\left (7 x^{2}+10 x +6\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.158 |
|
|
\[ {}3 x^{2} y^{\prime \prime }+2 x \left (-2 x^{2}+x +1\right ) y^{\prime }+\left (-8 x^{2}+2 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.078 |
|
|
\[ {}12 x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (3 x^{2}+35 x +11\right ) y^{\prime }-\left (-5 x^{2}-10 x +1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.676 |
|
|
\[ {}x^{2} \left (10 x^{2}+x +5\right ) y^{\prime \prime }+x \left (48 x^{2}+3 x +4\right ) y^{\prime }+\left (36 x^{2}+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.592 |
|
|
\[ {}18 x^{2} \left (1+x \right ) y^{\prime \prime }+3 x \left (x^{2}+11 x +5\right ) y^{\prime }-\left (-5 x^{2}-2 x +1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.235 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+x \left (2 x +3\right ) y^{\prime }-\left (1-x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.913 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+x \left (5+x \right ) y^{\prime }-\left (2-3 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.928 |
|
|
\[ {}3 x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.92 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-2 x +1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.823 |
|
|
\[ {}3 x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-\left (1+3 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.974 |
|
|
\[ {}2 x^{2} \left (x +3\right ) y^{\prime \prime }+x \left (1+5 x \right ) y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.184 |
|
|
\[ {}x^{2} \left (x +4\right ) y^{\prime \prime }-x \left (1-3 x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.994 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+5 x y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.798 |
|
|
\[ {}6 x^{2} y^{\prime \prime }+x \left (10-x \right ) y^{\prime }-\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.056 |
|
|
\[ {}x^{2} \left (3+4 x \right ) y^{\prime \prime }+x \left (11+4 x \right ) y^{\prime }-\left (3+4 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.063 |
|
|
\[ {}2 x^{2} \left (2+3 x \right ) y^{\prime \prime }+x \left (4+11 x \right ) y^{\prime }-\left (1-x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.658 |
|
|
\[ {}x^{2} \left (2+x \right ) y^{\prime \prime }+5 x \left (1-x \right ) y^{\prime }-\left (2-8 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.688 |
|
|
\[ {}8 x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+2 x \left (-13 x^{2}+1\right ) y^{\prime }+\left (-9 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.234 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-2 x \left (-x^{2}+2\right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.83 |
|
|
\[ {}x \left (x^{2}+3\right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }-8 y x = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.862 |
|
|
\[ {}4 x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+x \left (-19 x^{2}+7\right ) y^{\prime }-\left (14 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.367 |
|
|
\[ {}3 x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }+x \left (-11 x^{2}+1\right ) y^{\prime }+\left (-5 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.05 |
|
|
\[ {}2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }-x \left (-7 x^{2}+12\right ) y^{\prime }+\left (3 x^{2}+7\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.96 |
|
|
\[ {}2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+x \left (7 x^{2}+4\right ) y^{\prime }-\left (-3 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.013 |
|
|
\[ {}2 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+5 x \left (6 x^{2}+1\right ) y^{\prime }-\left (-40 x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.293 |
|
|
\[ {}x \left (x^{2}+1\right ) y^{\prime \prime }+\left (7 x^{2}+4\right ) y^{\prime }+8 y x = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.248 |
|
|
\[ {}2 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (8 x^{2}+3\right ) y^{\prime }-\left (-4 x^{2}+3\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.987 |
|
|
\[ {}9 x^{2} y^{\prime \prime }+3 x \left (x^{2}+3\right ) y^{\prime }-\left (-5 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.912 |
|
|
\[ {}6 x^{2} y^{\prime \prime }+x \left (6 x^{2}+1\right ) y^{\prime }+\left (9 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.023 |
|
|
\[ {}9 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+3 x \left (13 x^{2}+3\right ) y^{\prime }-\left (-25 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.037 |
|
|
\[ {}4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+4 x \left (6 x^{2}+1\right ) y^{\prime }-\left (-25 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.979 |
|
|
\[ {}8 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+2 x \left (34 x^{2}+5\right ) y^{\prime }-\left (-30 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.027 |
|
|
\[ {}2 x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (1-3 x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.945 |
|
|
\[ {}6 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+x \left (50 x^{2}+1\right ) y^{\prime }+\left (30 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.902 |
|
|
\[ {}28 x^{2} \left (1-3 x \right ) y^{\prime \prime }-7 x \left (5+9 x \right ) y^{\prime }+7 \left (2+9 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.357 |
|
|
\[ {}8 x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }+2 x \left (-21 x^{2}+10\right ) y^{\prime }-\left (35 x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.776 |
|
|
\[ {}4 x^{2} \left (x^{2}+3 x +1\right ) y^{\prime \prime }-4 x \left (-3 x^{2}-3 x +1\right ) y^{\prime }+3 \left (x^{2}-x +1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.905 |
|
|
\[ {}3 x^{2} \left (1+x \right )^{2} y^{\prime \prime }-x \left (-11 x^{2}-10 x +1\right ) y^{\prime }+\left (5 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.022 |
|
|
\[ {}4 x^{2} \left (x^{2}+2 x +3\right ) y^{\prime \prime }-x \left (-15 x^{2}-14 x +3\right ) y^{\prime }+\left (7 x^{2}+3\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.901 |
|
|
\[ {}x^{2} \left (x^{2}-2 x +1\right ) y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+\left (x +4\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.905 |
|
|
\[ {}2 x^{2} \left (2+x \right ) y^{\prime \prime }+5 x^{2} y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.016 |
|
|
\[ {}x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }-2 x \left (2 x^{2}+1\right ) y^{\prime }+\left (-2 x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.822 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (5-x \right ) y^{\prime }+\left (9-4 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.778 |
|
|
\[ {}4 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+12 x^{2} \left (1+x \right ) y^{\prime }+\left (3 x^{2}+3 x +1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.7 |
|
|
\[ {}x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }-x \left (-2 x^{2}-4 x +1\right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.655 |
|
|
\[ {}9 x^{2} y^{\prime \prime }+3 x \left (-2 x^{2}+3 x +5\right ) y^{\prime }+\left (-14 x^{2}+12 x +1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.41 |
|
|
\[ {}x^{2} \left (1+2 x \right ) y^{\prime \prime }+x \left (3 x^{2}+14 x +5\right ) y^{\prime }+\left (12 x^{2}+18 x +4\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.296 |
|
|
\[ {}16 x^{2} y^{\prime \prime }+4 x \left (2 x^{2}+x +6\right ) y^{\prime }+\left (18 x^{2}+5 x +1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.298 |
|
|
\[ {}9 x^{2} \left (1+x \right ) y^{\prime \prime }+3 x \left (-x^{2}+11 x +5\right ) y^{\prime }+\left (-7 x^{2}+16 x +1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.421 |
|
|
\[ {}36 x^{2} \left (-2 x +1\right ) y^{\prime \prime }+24 x \left (1-9 x \right ) y^{\prime }+\left (1-70 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.506 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (-x +3\right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.832 |
|
|
\[ {}x^{2} \left (-2 x +1\right ) y^{\prime \prime }-x \left (5-4 x \right ) y^{\prime }+\left (9-4 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.727 |
|
|
\[ {}2 x^{2} \left (2+x \right ) y^{\prime \prime }+x^{2} y^{\prime }+\left (1-x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.981 |
|
|
\[ {}2 x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (6-x \right ) y^{\prime }+\left (8-x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.99 |
|
|
\[ {}x^{2} \left (1+2 x \right ) y^{\prime \prime }+x \left (5+9 x \right ) y^{\prime }+\left (4+3 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.021 |
|
|
\[ {}x^{2} \left (-2 x +1\right ) y^{\prime \prime }-x \left (5+4 x \right ) y^{\prime }+\left (9+4 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.368 |
|
|
\[ {}x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (7+x \right ) y^{\prime }+\left (9-x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.914 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (-x^{2}+1\right ) y^{\prime }+\left (x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.691 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-3 x \left (-x^{2}+1\right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.716 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+2 x^{3} y^{\prime }+\left (3 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.731 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-2 x^{2}+1\right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.728 |
|
|
\[ {}2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+7 x^{3} y^{\prime }+\left (3 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.951 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-4 x^{2}+1\right ) y^{\prime }+\left (2 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.786 |
|
|
\[ {}4 x^{2} \left (x^{2}+4\right ) y^{\prime \prime }+3 x \left (3 x^{2}+8\right ) y^{\prime }+\left (-9 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.071 |
|
|
\[ {}3 x^{2} \left (x^{2}+3\right ) y^{\prime \prime }+x \left (11 x^{2}+3\right ) y^{\prime }+\left (5 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.68 |
|
|
\[ {}9 x^{2} y^{\prime \prime }-3 x \left (-2 x^{2}+7\right ) y^{\prime }+\left (2 x^{2}+25\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.793 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (-x^{2}+1\right ) y^{\prime }+\left (x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.713 |
|
|
\[ {}x^{2} \left (-2 x +1\right ) y^{\prime \prime }+3 x y^{\prime }+\left (1+4 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.886 |
|
|
\[ {}x \left (1+x \right ) y^{\prime \prime }+\left (1-x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.838 |
|
|
\[ {}x^{2} \left (1-x \right ) y^{\prime \prime }-x \left (3-5 x \right ) y^{\prime }+\left (4-5 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.749 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (9 x^{2}+1\right ) y^{\prime }+\left (25 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.77 |
|
|
\[ {}9 x^{2} y^{\prime \prime }+3 x \left (-x^{2}+1\right ) y^{\prime }+\left (7 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.946 |
|
|
\[ {}x \left (x^{2}+1\right ) y^{\prime \prime }+\left (-x^{2}+1\right ) y^{\prime }-8 y x = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.732 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+2 x \left (-x^{2}+4\right ) y^{\prime }+\left (7 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.839 |
|
|
\[ {}4 x^{2} \left (1+x \right ) y^{\prime \prime }+8 x^{2} y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.796 |
|
|
\[ {}9 x^{2} \left (x +3\right ) y^{\prime \prime }+3 x \left (3+7 x \right ) y^{\prime }+\left (3+4 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.995 |
|
|
\[ {}x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }-x \left (3 x^{2}+2\right ) y^{\prime }+\left (-x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.719 |
|
|
\[ {}16 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+8 x \left (9 x^{2}+1\right ) y^{\prime }+\left (49 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.923 |
|
|
\[ {}x^{2} \left (4+3 x \right ) y^{\prime \prime }-x \left (4-3 x \right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.174 |
|
|
\[ {}4 x^{2} \left (x^{2}+3 x +1\right ) y^{\prime \prime }+8 x^{2} \left (2 x +3\right ) y^{\prime }+\left (9 x^{2}+3 x +1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.863 |
|
|
\[ {}x^{2} \left (1-x \right )^{2} y^{\prime \prime }-x \left (-3 x^{2}+2 x +1\right ) y^{\prime }+\left (x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.829 |
|
|
\[ {}9 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+3 x \left (13 x^{2}+7 x +1\right ) y^{\prime }+\left (25 x^{2}+4 x +1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.873 |
|
|
\[ {}2 x^{2} \left (2+x \right ) y^{\prime \prime }-x \left (4-7 x \right ) y^{\prime }-\left (5-3 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.615 |
|
|
\[ {}x^{2} \left (-2 x +1\right ) y^{\prime \prime }+x \left (8-9 x \right ) y^{\prime }+\left (6-3 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.042 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (10 x^{2}+3\right ) y^{\prime }-\left (-14 x^{2}+15\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.549 |
|
|
\[ {}x^{2} \left (-2 x^{2}+1\right ) y^{\prime \prime }+x \left (-13 x^{2}+7\right ) y^{\prime }-14 x^{2} y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.079 |
|
|
\[ {}4 x^{2} \left (1+x \right ) y^{\prime \prime }+4 x \left (1+2 x \right ) y^{\prime }-\left (1+3 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.484 |
|
|
\[ {}2 x^{2} \left (2+3 x \right ) y^{\prime \prime }+x \left (4+21 x \right ) y^{\prime }-\left (1-9 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.562 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (2+x \right ) y^{\prime }-\left (2-3 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.276 |
|
|
\[ {}4 x^{2} \left (1+x \right ) y^{\prime \prime }+4 x \left (3+8 x \right ) y^{\prime }-\left (5-49 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.565 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (3+10 x \right ) y^{\prime }+30 y x = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.428 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-3 \left (x +3\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.306 |
|
|
\[ {}x^{2} \left (1+2 x \right ) y^{\prime \prime }+x \left (9+13 x \right ) y^{\prime }+\left (7+5 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.225 |
|
|
\[ {}4 x^{2} \left (1+2 x \right ) y^{\prime \prime }-2 x \left (4-x \right ) y^{\prime }-\left (7+5 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.398 |
|
|
\[ {}3 x^{2} \left (x +3\right ) y^{\prime \prime }-x \left (15+x \right ) y^{\prime }-20 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.127 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (1-10 x \right ) y^{\prime }-\left (9-10 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.098 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+3 x^{2} y^{\prime }-\left (6-x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.904 |
|
|
\[ {}x^{2} \left (1+2 x \right ) y^{\prime \prime }-2 x \left (3+14 x \right ) y^{\prime }+\left (6+100 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.873 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (6+11 x \right ) y^{\prime }+\left (6+32 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.921 |
|
|
\[ {}4 x^{2} \left (1+x \right ) y^{\prime \prime }+4 x \left (1+4 x \right ) y^{\prime }-\left (49+27 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.087 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-2 x^{2}+7\right ) y^{\prime }+12 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.278 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (-x^{2}+7\right ) y^{\prime }+12 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.098 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (2 x^{2}+1\right ) y^{\prime }-\left (-10 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.824 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (-2 x^{2}+1\right ) y^{\prime }-4 \left (2 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.23 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (-3 x^{2}+1\right ) y^{\prime }-4 \left (-3 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.392 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (11 x^{2}+5\right ) y^{\prime }+24 x^{2} y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.879 |
|
|
\[ {}4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+8 x y^{\prime }-\left (-x^{2}+35\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.934 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-x^{2}+5\right ) y^{\prime }-\left (25 x^{2}+7\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.404 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (2 x^{2}+5\right ) y^{\prime }-21 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.219 |
|
|
\[ {}4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+4 x \left (x^{2}+2\right ) y^{\prime }-\left (x^{2}+15\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.877 |
|
|
\[ {}y^{\prime \prime }-\frac {2 \left (1+t \right ) y^{\prime }}{t^{2}+2 t -1}+\frac {2 y}{t^{2}+2 t -1} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.977 |
|
|
\[ {}y^{\prime \prime }-4 t y^{\prime }+\left (4 t^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.699 |
|
|
\[ {}\left (-t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[_Gegenbauer] |
✓ |
✓ |
1.0 |
|
|
\[ {}\left (t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.766 |
|
|
\[ {}\left (-t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.681 |
|
|
\[ {}\left (2 t +1\right ) y^{\prime \prime }-4 \left (1+t \right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.935 |
|
|
\[ {}t^{2} y^{\prime \prime }+t y^{\prime }+\left (t^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.823 |
|
|
\[ {}y^{\prime \prime }-\frac {2 t y^{\prime }}{t^{2}+1}+\frac {2 y}{t^{2}+1} = t^{2}+1 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.499 |
|
|
\[ {}y^{\prime \prime }+\left (t^{2}+2 t +1\right ) y^{\prime }-\left (4+4 t \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.904 |
|
|
\[ {}\left ({\mathrm e}^{t}-1\right ) y^{\prime \prime }+{\mathrm e}^{t} y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.799 |
|
|
\[ {}2 t^{2} y^{\prime \prime }+3 t y^{\prime }-\left (1+t \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.257 |
|
|
\[ {}2 t y^{\prime \prime }+\left (1-2 t \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[_Laguerre] |
✓ |
✓ |
0.927 |
|
|
\[ {}2 t y^{\prime \prime }+\left (1+t \right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.005 |
|
|
\[ {}2 t^{2} y^{\prime \prime }-t y^{\prime }+\left (1+t \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.835 |
|
|
\[ {}2 t^{2} y^{\prime \prime }+\left (t^{2}-t \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.894 |
|
|
\[ {}t^{2} y^{\prime \prime }+\left (-t^{2}+t \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.137 |
|
|
\[ {}t y^{\prime \prime }-\left (t^{2}+2\right ) y^{\prime }+t y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Lienard] |
✓ |
✓ |
0.752 |
|
|
\[ {}t^{2} y^{\prime \prime }+t \left (1+t \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.762 |
|
|
\[ {}t y^{\prime \prime }-\left (4+t \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Laguerre] |
✓ |
✓ |
0.829 |
|
|
\[ {}t^{2} y^{\prime \prime }+\left (t^{2}-3 t \right ) y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.205 |
|
|
\[ {}t y^{\prime \prime }+t y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.207 |
|
|
\[ {}t y^{\prime \prime }+\left (-t^{2}+1\right ) y^{\prime }+4 t y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.794 |
|
|
\[ {}t^{2} y^{\prime \prime }-t \left (1+t \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.724 |
|
|
\[ {}y-x^{2} \sqrt {x^{2}-y^{2}}-x y^{\prime } = 0 \] |
1 |
0 |
0 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.19 |
|
|
\[ {}x^{2} \left (x +4\right ) y^{\prime \prime }+7 x y^{\prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.237 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+\left (-x^{2}+x \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.909 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+5 x y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.83 |
|
|
\[ {}\left (x^{3}+2 x^{2}\right ) y^{\prime \prime }-x y^{\prime }+\left (1-x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.174 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-3 \left (x^{2}+x \right ) y^{\prime }+\left (2+3 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.963 |
|
|
\[ {}4 x^{2} \left (1-x \right ) y^{\prime \prime }+3 x \left (1+2 x \right ) y^{\prime }-3 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.023 |
|
|
\[ {}2 x^{2} \left (1-3 x \right ) y^{\prime \prime }+5 x y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.461 |
|
|
\[ {}4 x^{2} \left (1+x \right ) y^{\prime \prime }-5 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.056 |
|
|
\[ {}x^{2} \left (x +4\right ) y^{\prime \prime }+x \left (-1+x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.037 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (1+x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.704 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (2 x +3\right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.753 |
|
|
\[ {}x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }-5 x y^{\prime }+9 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.906 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x^{2}-1\right ) y^{\prime }+\left (-x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.738 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (2 x -1\right ) y^{\prime }+x \left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.906 |
|
|
\[ {}x^{2} y^{\prime \prime }-x^{2} y^{\prime }+\left (x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.365 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x^{2} y^{\prime }-\left (3 x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.876 |
|
|
\[ {}x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-9 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.925 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x -7\right ) y^{\prime }+\left (x +12\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.297 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (x -4\right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.348 |
|
|
\[ {}\left (-2 x +1\right ) y^{\prime \prime }+4 x y^{\prime }-4 y = x^{2}-x \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.76 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+\left (-x^{2}+x \right ) y^{\prime }-y = x^{3}+1 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.781 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 6 \left (-x^{2}+1\right )^{2} \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.307 |
|
|
\[ {}\left (x^{2}+2 x \right ) y^{\prime \prime }-\left (2 x +2\right ) y^{\prime }+2 y = x^{2} \left (2+x \right )^{2} \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.922 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+5 x y^{\prime }+\left (1+x \right ) y = x \left (x^{2}+x +1\right ) \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.786 |
|
|
\[ {}\left (x^{3}+2 x^{2}\right ) y^{\prime \prime }-x y^{\prime }+\left (1-x \right ) y = x^{2} \left (1+x \right )^{2} \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.04 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+6\right ) y = {\mathrm e}^{-x^{2}} \sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.974 |
|
|
\[ {}\left (-z^{2}+1\right ) y^{\prime \prime }-3 y^{\prime } z +\lambda y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer] |
✓ |
✓ |
0.874 |
|
|
\[ {}4 z y^{\prime \prime }+2 \left (1-z \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.306 |
|
|
\[ {}f^{\prime \prime }+2 \left (z -1\right ) f^{\prime }+4 f = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.558 |
|
|
\[ {}z^{2} y^{\prime \prime }-\frac {3 y^{\prime } z}{2}+\left (z +1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.918 |
|
|
\[ {}z y^{\prime \prime }-2 y^{\prime }+z y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Lienard] |
✓ |
✓ |
0.721 |
|
|
\[ {}z y^{\prime \prime }+\left (2 z -3\right ) y^{\prime }+\frac {4 y}{z} = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.768 |
|
|
\[ {}y^{\prime } = \frac {1-y^{2}}{2+2 y x} \] |
1 |
1 |
1 |
exact |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.908 |
|
|
\[ {}x y^{\prime \prime }+\left (-2 x +1\right ) y^{\prime }+\left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.338 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.386 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[_Gegenbauer] |
✓ |
✓ |
0.376 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}-1\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.766 |
|
|
\[ {}x y^{\prime \prime }-\left (1+2 x \right ) y^{\prime }+2 y = 8 x^{2} {\mathrm e}^{2 x} \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.275 |
|
|
\[ {}y^{\prime \prime }+2 x y^{\prime }+4 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_erf] |
✓ |
✓ |
0.76 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+3 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.514 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }-3 y x = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.47 |
|
|
\[ {}\left (-4 x^{2}+1\right ) y^{\prime \prime }-20 x y^{\prime }-16 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer] |
✓ |
✓ |
0.713 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-6 x y^{\prime }+12 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer] |
✓ |
✓ |
0.494 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+\left (2+x \right ) y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.524 |
|
|
\[ {}\left (2 x^{2}+1\right ) y^{\prime \prime }+7 x y^{\prime }+2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.417 |
|
|
\[ {}4 y^{\prime \prime }+x y^{\prime }+4 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Lienard] |
✓ |
✓ |
1.167 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }-4 y = 6 \,{\mathrm e}^{x} \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.307 |
|
|
\[ {}4 x y^{\prime \prime }-x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.005 |
|
|
\[ {}6 x^{2} y^{\prime \prime }+x \left (1+18 x \right ) y^{\prime }+\left (1+12 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.057 |
|
|
\[ {}3 x^{2} y^{\prime \prime }-x \left (x +8\right ) y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.079 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x \left (1+2 x \right ) y^{\prime }+2 \left (4 x -1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.004 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x^{2} y^{\prime }+\left (1+2 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.831 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (3-2 x \right ) y^{\prime }+\left (-2 x +1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.834 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+\left (4-x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.828 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (-x +3\right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.297 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (2 \sqrt {5}-1\right ) x y^{\prime }+\left (\frac {19}{4}-3 x^{2}\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.399 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x -3\right ) y^{\prime }+\left (4-x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.793 |
|
|
\[ {}x^{2} y^{\prime \prime }+x^{2} y^{\prime }-\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.868 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x^{2} y^{\prime }+\left (x -\frac {3}{4}\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.343 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+x^{2} y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.471 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x^{2}+6\right ) y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.46 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.317 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.769 |
|
|
\[ {}x^{2} y^{\prime \prime }-x^{2} y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.736 |
|
|
\[ {}x^{2} y^{\prime \prime }-x^{2} y^{\prime }-\left (2+3 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.195 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (5-x \right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.74 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x \left (1-x \right ) y^{\prime }+\left (2 x -9\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.822 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x \left (2+x \right ) y^{\prime }+2 \left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.009 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (1-x \right ) y^{\prime }+\left (1-x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.716 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x \left (1+2 x \right ) y^{\prime }+\left (4 x -1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.847 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x +4\right ) y^{\prime }+\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.533 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {9}{4}\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.843 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+y x = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Lienard] |
✓ |
✓ |
0.746 |
|
|
\[ {}2 x y^{\prime \prime }+5 \left (-2 x +1\right ) y^{\prime }-5 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.049 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.266 |
|
|
\[ {}x^{2} y^{\prime \prime }+\frac {3 x y^{\prime }}{2}-\frac {\left (1+x \right ) y}{2} = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.335 |
|
|
\[ {}y^{\prime } = \cos \left (2 x \right )+\left (\sin \left (2 x \right )+y\right ) y \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.217 |
|
|
\[ {}y^{\prime } = 1-x -x^{3}+\left (2 x^{2}+1\right ) y-x y^{2} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_Riccati] |
✓ |
✓ |
1.539 |
|
|
\[ {}y^{\prime } = x \left (2+x^{2} y-y^{2}\right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
0.938 |
|
|
\[ {}y^{\prime }+4 \csc \left (x \right ) = \left (3-\cot \left (x \right )\right ) y+y^{2} \sin \left (x \right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.747 |
|
|
\[ {}x y^{\prime }-y+y^{2} = x^{\frac {2}{3}} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
11.781 |
|
|
\[ {}x y^{\prime } = y+x \sqrt {x^{2}+y^{2}} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.04 |
|
|
\[ {}\left (x -a \right ) \left (-b +x \right ) y^{\prime }+k \left (x +y-a \right ) \left (x +y-b \right )+y^{2} = 0 \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
1.938 |
|
|
\[ {}x \left (-x^{3}+1\right ) y^{\prime } = x^{2}+\left (1-2 y x \right ) y \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.166 |
|
|
\[ {}\left (c \,x^{2}+b x +a \right )^{2} \left (y^{2}+y^{\prime }\right )+A = 0 \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_rational, _Riccati] |
✓ |
✓ |
3.106 |
|
|
\[ {}\left (2+3 x -y x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.97 |
|
|
\[ {}\left (x^{2} a^{2}+\left (x^{2}+y^{2}\right )^{2}\right ) y^{\prime } = a^{2} x y \] |
1 |
1 |
4 |
first_order_ode_lie_symmetry_calculated |
[_rational] |
✓ |
✓ |
2.498 |
|
|
\[ {}x y^{\prime }-a y+y^{2} = x^{-2 a} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
0.217 |
|
|
\[ {}x y^{\prime }-a y+y^{2} = x^{-\frac {2 a}{3}} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.683 |
|
|
\[ {}x y^{\prime \prime }+\left (x +n \right ) y^{\prime }+\left (n +1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.809 |
|
|
\[ {}x^{4} y^{\prime \prime }+x y^{\prime }+y = 0 \] |
1 |
0 |
0 |
second order series method. Irregular singular point |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.415 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (2 x^{2}+x \right ) y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.881 |
|
|
\[ {}\left (4 x^{3}-14 x^{2}-2 x \right ) y^{\prime \prime }-\left (6 x^{2}-7 x +1\right ) y^{\prime }+\left (6 x -1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.116 |
|
|
\[ {}x^{2} y^{\prime \prime }+x^{2} y^{\prime }+\left (x -2\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.833 |
|
|
\[ {}x^{2} y^{\prime \prime }-x^{2} y^{\prime }+\left (x -2\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.793 |
|
|
\[ {}x^{2} \left (1-4 x \right ) y^{\prime \prime }+\left (\left (1-n \right ) x -\left (6-4 n \right ) x^{2}\right ) y^{\prime }+n \left (1-n \right ) x y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.455 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }+\left (x -9\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.021 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }+\left (3 x -1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.419 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (x^{2}+4 x \right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.307 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-\left (2+3 x \right ) y^{\prime }+\frac {\left (2 x -1\right ) y}{x} = \sqrt {x} \] |
1 |
0 |
0 |
second order series method. Irregular singular point |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
0.521 |
|
|
\[ {}x \left (1-x \right ) y^{\prime \prime }+\left (\frac {3}{2}-2 x \right ) y^{\prime }-\frac {y}{4} = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[_Jacobi] |
✓ |
✓ |
0.981 |
|
|
\[ {}2 x \left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.099 |
|
|
\[ {}2 x \left (1-x \right ) y^{\prime \prime }+\left (1-11 x \right ) y^{\prime }-10 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[_Jacobi] |
✓ |
✓ |
0.976 |
|
|
\[ {}x \left (1-x \right ) y^{\prime \prime }+\frac {\left (-2 x +1\right ) y^{\prime }}{3}+\frac {20 y}{9} = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[_Jacobi] |
✓ |
✓ |
1.019 |
|
|
\[ {}4 y^{\prime \prime }+\frac {3 \left (-x^{2}+2\right ) y}{\left (-x^{2}+1\right )^{2}} = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.851 |
|
|
\[ {}u^{\prime \prime }-\frac {2 u^{\prime }}{x}-a^{2} u = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.504 |
|
|
\[ {}u^{\prime \prime }+\frac {2 u^{\prime }}{x}-a^{2} u = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.842 |
|
|
\[ {}u^{\prime \prime }+\frac {2 u^{\prime }}{x}+a^{2} u = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.53 |
|
|
\[ {}u^{\prime \prime }+\frac {4 u^{\prime }}{x}-a^{2} u = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.526 |
|
|
\[ {}u^{\prime \prime }+\frac {4 u^{\prime }}{x}+a^{2} u = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.727 |
|
|
\[ {}y^{\prime \prime }-a^{2} y = \frac {6 y}{x^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.542 |
|
|
\[ {}y^{\prime \prime }+n^{2} y = \frac {6 y}{x^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.81 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-\left (x^{2}+\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.51 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\frac {\left (-9 a^{2}+4 x^{2}\right ) y}{4 a^{2}} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.871 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {25}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.904 |
|
|
\[ {}y^{\prime \prime }+q y^{\prime } = \frac {2 y}{x^{2}} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.484 |
|
|
\[ {}x y^{\prime \prime }+3 y^{\prime }+4 x^{3} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.524 |
|
|
\[ {}x^{2} \left (2-x \right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.369 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.141 |
|
|
\[ {}x y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.377 |
|
|
\[ {}3 x y^{\prime \prime }-2 \left (3 x -1\right ) y^{\prime }+\left (-2+3 x \right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.454 |
|
|
\[ {}x \left (1+x \right ) y^{\prime \prime }-\left (-1+x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.799 |
|
|
\[ {}\left (x^{2}+2 x \right ) y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.953 |
|
|
\[ {}\left (x^{2}+2 x \right ) y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.154 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.586 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.801 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.544 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.317 |
|
|
\[ {}\left (2 x -3\right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.789 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-3 y = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Hermite] |
✓ |
✓ |
0.701 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+y = \cos \left (x \right ) \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.765 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+2 y = \cos \left (x \right ) \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.111 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime }+y = \tan \left (x \right ) \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.801 |
|
|
\[ {}x \left (1+x \right )^{2} y^{\prime \prime }+\left (-x^{2}+1\right ) y^{\prime }+\left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.509 |
|
|
\[ {}2 x y^{\prime \prime }-y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.736 |
|
|
\[ {}x y^{\prime \prime }+x y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.302 |
|
|
\[ {}x \left (-1+x \right )^{2} y^{\prime \prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.848 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+x^{2} y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.308 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.919 |
|
|
\[ {}y+\left (y x +x -3 y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.996 |
|
|
\[ {}x y^{\prime \prime }-\left (2+x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[_Laguerre] |
✓ |
✓ |
1.443 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 2 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.29 |
|
|
\[ {}\left (x^{2}+4\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 8 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.045 |
|
|
\[ {}\left (1+x \right ) y^{\prime \prime }-\left (2 x +3\right ) y^{\prime }+\left (2+x \right ) y = \left (x^{2}+2 x +1\right ) {\mathrm e}^{2 x} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.391 |
|
|
\[ {}y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime }-10 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.721 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (2 x +3\right ) y^{\prime }+\left (x^{2}+3 x +3\right ) y = \left (-x^{2}+6\right ) {\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.4 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x^{3} y^{\prime }+\left (x^{2}+1\right )^{2} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.576 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (-4 x^{2}+x \right ) y^{\prime }+\left (4 x^{2}-2 x +1\right ) y = \left (x^{2}-x +1\right ) {\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.043 |
|
|
|
|||||||||
|
\[ {}\left (1+x \right ) y^{\prime \prime }-\left (4+3 x \right ) y^{\prime }+3 y = \left (2+3 x \right ) {\mathrm e}^{3 x} \] |
1 |
1 |
1 |
kovacic, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.793 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (9 x^{2}+6\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.708 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+4 y x = 4 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.74 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = \frac {-x^{2}+1}{x} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.378 |
|
|
\[ {}\left (2 x^{3}-1\right ) y^{\prime \prime \prime }-6 x^{2} y^{\prime \prime }+6 x y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
2.203 |
|
|
\[ {}2 \left (x^{3}+x^{2}\right ) y^{\prime \prime }-\left (-3 x^{2}+x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.957 |
|
|
\[ {}4 x y^{\prime \prime }+2 \left (1-x \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.851 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+y x = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Lienard] |
✓ |
✓ |
0.763 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.962 |
|
|
\[ {}2 x^{3} y^{\prime \prime }+x^{2} y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.214 |
|
|
\[ {}x^{3} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.642 |
|
|
\[ {}x \left (-x^{2}+2\right ) y^{\prime \prime }-\left (x^{2}+4 x +2\right ) \left (\left (1-x \right ) y^{\prime }+y\right ) = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.243 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }-\left (1+2 x \right ) \left (-y+x y^{\prime }\right ) = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.987 |
|
|
\[ {}2 x^{2} \left (2-x \right ) y^{\prime \prime }-x \left (4-x \right ) y^{\prime }+\left (-x +3\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.02 |
|
|
\[ {}x^{2} \left (1-x \right ) y^{\prime \prime }+\left (5 x -4\right ) x y^{\prime }+\left (6-9 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.326 |
|
|
\[ {}x y^{\prime \prime }+\left (4 x^{2}+1\right ) y^{\prime }+4 x \left (x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.806 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+8 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.019 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+8 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.874 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+12 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer] |
✓ |
✓ |
1.379 |
|
|
\[ {}x \left (2+x \right ) y^{\prime \prime }+2 \left (1+x \right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.924 |
|
|
\[ {}\sin \left (x \right ) y^{\prime \prime }-2 \cos \left (x \right ) y^{\prime }-\sin \left (x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.526 |
|
|
\[ {}x \left (2+x \right ) y^{\prime \prime }+\left (1+x \right ) y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.027 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.642 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.257 |
|
|
\[ {}\left (x^{2}-2 x +10\right ) y^{\prime \prime }+x y^{\prime }-4 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.867 |
|
|
\[ {}\left (x^{2}-2 x +10\right ) y^{\prime \prime }+x y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.693 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Hermite] |
✓ |
✓ |
0.572 |
|
|
\[ {}\left (2+x \right ) y^{\prime \prime }+x y^{\prime }-y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.682 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-6 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.3 |
|
|
\[ {}\left (x^{2}+2\right ) y^{\prime \prime }+3 x y^{\prime }-y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.431 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.371 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+8 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.85 |
|
|
\[ {}y^{\prime \prime }+{\mathrm e}^{x} y^{\prime }-y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.549 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (\frac {5}{3} x +x^{2}\right ) y^{\prime }-\frac {y}{3} = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.193 |
|
|
\[ {}2 x y^{\prime \prime }-y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.878 |
|
|
\[ {}2 x y^{\prime \prime }-\left (2 x +3\right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[_Laguerre] |
✓ |
✓ |
1.003 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+3 x y^{\prime }+\left (2 x -1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.033 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.69 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.74 |
|
|
\[ {}x y^{\prime \prime }+\left (x -6\right ) y^{\prime }-3 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.184 |
|
|
\[ {}x^{4} y^{\prime \prime }+\lambda y = 0 \] |
1 |
0 |
0 |
second order series method. Irregular singular point |
[[_Emden, _Fowler]] |
✗ |
N/A |
0.128 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}-25\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.892 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (36 x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.835 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.744 |
|
|
\[ {}x y^{\prime \prime }+3 y^{\prime }+x^{3} y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.741 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.93 |
|
|
\[ {}16 x^{2} y^{\prime \prime }+32 x y^{\prime }+\left (x^{4}-12\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.777 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }+y x = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.564 |
|
|
\[ {}x y^{\prime \prime }-\left (2+x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Laguerre] |
✓ |
✓ |
0.821 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.22 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer] |
✓ |
✓ |
0.658 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.52 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+30 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer] |
✓ |
✓ |
1.342 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+y x = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Lienard] |
✓ |
✓ |
0.817 |
|
|
\[ {}x y^{\prime \prime }+\left (1+2 x \right ) y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.855 |
|
|
\[ {}2 x \left (-1+x \right ) y^{\prime \prime }-\left (1+x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[_Jacobi] |
✓ |
✓ |
0.947 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+4 y x = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.162 |
|
|
\[ {}x y^{\prime \prime }+\left (2-2 x \right ) y^{\prime }+\left (x -2\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.828 |
|
|
\[ {}x^{2} y^{\prime \prime }+6 x y^{\prime }+\left (4 x^{2}+6\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.784 |
|
|
\[ {}x y^{\prime \prime }+\left (-2 x +1\right ) y^{\prime }+\left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.873 |
|
|
\[ {}x \left (1-x \right ) y^{\prime \prime }+\left (\frac {1}{2}+2 x \right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[_Jacobi] |
✓ |
✓ |
0.993 |
|
|
\[ {}4 \left (t^{2}-3 t +2\right ) y^{\prime \prime }-2 y^{\prime }+y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.936 |
|
|
\[ {}2 \left (t^{2}-5 t +6\right ) y^{\prime \prime }+\left (2 t -3\right ) y^{\prime }-8 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.929 |
|
|
\[ {}3 t \left (1+t \right ) y^{\prime \prime }+t y^{\prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.435 |
|
|
\[ {}x^{2} y^{\prime \prime }+\frac {\left (x +\frac {3}{4}\right ) y}{4} = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.842 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\frac {\left (x^{2}-1\right ) y}{4} = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.826 |
|
|
\[ {}x y^{\prime \prime }+\left (-2 x +1\right ) y^{\prime }+\left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.88 |
|
|
\[ {}x y^{\prime \prime }-\left (1+x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Laguerre] |
✓ |
✓ |
0.82 |
|
|
\[ {}x y^{\prime \prime }+3 y^{\prime }+4 x^{3} y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.762 |
|
|
\[ {}x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+2 x \left (-x^{2}+1\right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.609 |
|
|
\[ {}2 x y^{\prime \prime }+\left (x -2\right ) y^{\prime }-y = x^{2}-1 \] |
1 |
1 |
1 |
kovacic, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.844 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+y x = \sec \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.996 |
|
|
\[ {}y^{\prime \prime }+2 x^{2} y^{\prime }+\left (x^{4}+2 x -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.813 |
|
|
\[ {}\sin \left (x \right ) u^{\prime \prime }+2 \cos \left (x \right ) u^{\prime }+\sin \left (x \right ) u = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[_Lienard] |
✓ |
✓ |
0.459 |
|
|
\[ {}u^{\prime \prime }-\left (1+2 x \right ) u^{\prime }+\left (x^{2}+x -1\right ) u = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.458 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+\left (1+\frac {2}{\left (1+3 x \right )^{2}}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.759 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.888 |
|
|
\[ {}y^{\prime \prime }+\frac {2 y^{\prime }}{x}-\frac {2 y}{\left (1+x \right )^{2}} = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.034 |
|
|
\[ {}\left (\phi ^{\prime }-\frac {\phi ^{2}}{2}\right ) \sin \left (\theta \right )^{2}-\phi \sin \left (\theta \right ) \cos \left (\theta \right ) = \frac {\cos \left (2 \theta \right )}{2}+1 \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
3.218 |
|
|
\[ {}y^{\prime \prime }-\frac {y^{\prime }}{\sqrt {x}}+\frac {\left (x +\sqrt {x}-8\right ) y}{4 x^{2}} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.381 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.289 |
|
|
\[ {}x y^{\prime \prime }-\left (1+x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[_Laguerre] |
✓ |
✓ |
0.32 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[_Gegenbauer] |
✓ |
✓ |
0.384 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.239 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.056 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Hermite] |
✓ |
✓ |
0.426 |
|
|
\[ {}y^{\prime \prime }+\left (-1+x \right )^{2} y^{\prime }-\left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.944 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.513 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[_Gegenbauer] |
✓ |
✓ |
1.098 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.94 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x^{2} y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.03 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x \left (1+x \right ) y^{\prime }+2 \left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.069 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer] |
✓ |
✓ |
1.139 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = \left (x^{2}-1\right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.854 |
|
|
\[ {}\left (x^{2}+x \right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }-\left (2+x \right ) y = x \left (1+x \right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.669 |
|
|
\[ {}\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = \left (1-x \right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.777 |
|
|
\[ {}x y^{\prime \prime }-\left (1+x \right ) y^{\prime }+y = x^{2} {\mathrm e}^{2 x} \] |
1 |
1 |
1 |
kovacic, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.565 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[_Gegenbauer] |
✓ |
✓ |
0.33 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.349 |
|
|
\[ {}y^{\prime \prime }-\frac {x y^{\prime }}{-1+x}+\frac {y}{-1+x} = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.333 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (2+x \right ) y^{\prime }+\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.334 |
|
|
\[ {}x y^{\prime \prime }-\left (1+2 x \right ) y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.34 |
|
|
\[ {}y^{\prime \prime }+\left (1+x \right ) y^{\prime }-y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.735 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.414 |
|
|
\[ {}2 x y^{\prime \prime }+\left (1+x \right ) y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.396 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+x y^{\prime }-\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.251 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-8 x^{2} y^{\prime }+\left (4 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.983 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+y x = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Lienard] |
✓ |
✓ |
0.831 |
|
|
\[ {}x^{2} y^{\prime \prime }-x^{2} y^{\prime }+\left (x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.174 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.476 |
|
|
\[ {}x \left (1-x \right ) y^{\prime \prime }+\left (\frac {3}{2}-2 x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[_Jacobi] |
✓ |
✓ |
1.209 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+\left (5 x +4\right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.381 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+y = x \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.697 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.394 |
|
|
\[ {}y^{\prime \prime }-\left (1+x \right ) y^{\prime }-y x = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.73 |
|
|
\[ {}x y^{\prime \prime }-4 y^{\prime }+y x = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Lienard] |
✓ |
✓ |
0.938 |
|
|
\[ {}2 x y^{\prime \prime }+\left (1-x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.296 |
|
|
\[ {}x y^{\prime \prime }-\left (-1+x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[_Laguerre] |
✓ |
✓ |
0.938 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Hermite] |
✓ |
✓ |
0.621 |
|
|
\[ {}\left (2+x \right ) y^{\prime \prime }+x y^{\prime }-y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.733 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-6 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.367 |
|
|
\[ {}\left (x^{2}+2\right ) y^{\prime \prime }+3 x y^{\prime }-y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.019 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.391 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+8 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.01 |
|
|
\[ {}y^{\prime \prime }+{\mathrm e}^{x} y^{\prime }-y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.808 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (\frac {5}{3} x +x^{2}\right ) y^{\prime }-\frac {y}{3} = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.374 |
|
|
\[ {}2 x y^{\prime \prime }-y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.217 |
|
|
\[ {}2 x y^{\prime \prime }-\left (2 x +3\right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[_Laguerre] |
✓ |
✓ |
1.201 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+3 x y^{\prime }+\left (2 x -1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.293 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.808 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.863 |
|
|
\[ {}x y^{\prime \prime }+\left (x -6\right ) y^{\prime }-3 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.065 |
|
|
\[ {}y^{\prime \prime }+\frac {2 y^{\prime }}{t}+\lambda y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.817 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}-25\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.764 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (36 x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.837 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.532 |
|
|
\[ {}x y^{\prime \prime }+3 y^{\prime }+x^{3} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.556 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.819 |
|
|
\[ {}16 x^{2} y^{\prime \prime }+32 x y^{\prime }+\left (x^{4}-12\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.698 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (16 x^{2}+3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.516 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }+y x = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.704 |
|
|
\[ {}x y^{\prime \prime }-\left (2+x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Laguerre] |
✓ |
✓ |
0.9 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.344 |
|
|
\[ {}2 y^{\prime \prime }+t y^{\prime }-2 y = 10 \] |
1 |
0 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.176 |
|
|
\[ {}y^{\prime \prime } = 2 x +\left (x^{2}-y^{\prime }\right )^{2} \] |
1 |
1 |
1 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.482 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.348 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+10 x y^{\prime }+20 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.539 |
|
|
\[ {}\left (x^{2}+4\right ) y^{\prime \prime }+2 x y^{\prime }-12 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.497 |
|
|
\[ {}\left (x^{2}-9\right ) y^{\prime \prime }+3 x y^{\prime }-3 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.881 |
|
|
\[ {}\left (2 x^{2}+1\right ) y^{\prime \prime }-5 x y^{\prime }+3 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.726 |
|
|
\[ {}\left (-4 x^{2}+1\right ) y^{\prime \prime }+6 x y^{\prime }-4 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer] |
✓ |
✓ |
0.975 |
|
|
\[ {}\left (2 x^{2}+1\right ) y^{\prime \prime }+3 x y^{\prime }-3 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.517 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+3 y = x^{2} \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.54 |
|
|
\[ {}2 y^{\prime \prime }+9 x y^{\prime }-36 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.346 |
|
|
\[ {}\left (x^{2}+4\right ) y^{\prime \prime }+3 x y^{\prime }-8 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.546 |
|
|
\[ {}\left (2 x^{2}+1\right ) y^{\prime \prime }+11 x y^{\prime }+9 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.729 |
|
|
\[ {}\left (x^{2}-2 x +2\right ) y^{\prime \prime }-4 \left (-1+x \right ) y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.398 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}-1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.984 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }-\left (4 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.918 |
|
|
\[ {}2 x^{2} \left (1-x \right ) y^{\prime \prime }-x \left (1+7 x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.114 |
|
|
\[ {}2 x y^{\prime \prime }+5 \left (-2 x +1\right ) y^{\prime }-5 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.259 |
|
|
\[ {}2 x y^{\prime \prime }+\left (2-x \right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.936 |
|
|
\[ {}2 x \left (x +3\right ) y^{\prime \prime }-3 \left (1+x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.13 |
|
|
\[ {}2 x y^{\prime \prime }+\left (-2 x^{2}+1\right ) y^{\prime }-4 y x = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.045 |
|
|
\[ {}x \left (4-x \right ) y^{\prime \prime }+\left (2-x \right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.237 |
|
|
\[ {}2 x y^{\prime \prime }+\left (1+2 x \right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.277 |
|
|
\[ {}2 x y^{\prime \prime }+\left (1+2 x \right ) y^{\prime }-5 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.281 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-3 x \left (1-x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.813 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+x \left (4 x -1\right ) y^{\prime }+2 \left (3 x -1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.089 |
|
|
\[ {}2 x y^{\prime \prime }-\left (2 x^{2}+1\right ) y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.943 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (1+x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.939 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x -3\right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.887 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (1+3 x \right ) y^{\prime }+\left (1-6 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.517 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (-1+x \right ) y^{\prime }+\left (1-x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.836 |
|
|
\[ {}x \left (x -2\right ) y^{\prime \prime }+2 \left (-1+x \right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.927 |
|
|
\[ {}x \left (x -2\right ) y^{\prime \prime }+2 \left (-1+x \right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.97 |
|
|
\[ {}4 \left (x -4\right )^{2} y^{\prime \prime }+\left (x -4\right ) \left (x -8\right ) y^{\prime }+y x = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.074 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x \left (1+x \right ) y^{\prime }+\left (1-3 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.995 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x \left (x -2\right ) y^{\prime }+2 \left (2-3 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.023 |
|
|
\[ {}x^{2} \left (1+2 x \right ) y^{\prime \prime }+2 x \left (6 x +1\right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.134 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (2+3 x \right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.899 |
|
|
\[ {}x y^{\prime \prime }-\left (x +3\right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Laguerre] |
✓ |
✓ |
1.056 |
|
|
\[ {}x \left (1+x \right ) y^{\prime \prime }+\left (5+x \right ) y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.198 |
|
|
\[ {}x \left (1+x \right ) y^{\prime \prime }+\left (5+x \right ) y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.197 |
|
|
\[ {}x^{2} y^{\prime \prime }+x^{2} y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.913 |
|
|
\[ {}x y^{\prime \prime }-2 \left (2+x \right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.594 |
|
|
\[ {}x y^{\prime \prime }+\left (2 x +3\right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.973 |
|
|
\[ {}x \left (x +3\right ) y^{\prime \prime }-9 y^{\prime }-6 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.073 |
|
|
\[ {}x \left (-2 x +1\right ) y^{\prime \prime }-2 \left (2+x \right ) y^{\prime }+8 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.151 |
|
|
\[ {}x y^{\prime \prime }+\left (x^{3}-1\right ) y^{\prime }+x^{2} y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.916 |
|
|
\[ {}x^{2} \left (4 x -1\right ) y^{\prime \prime }+x \left (1+5 x \right ) y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.336 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+2 x \left (2-x \right ) y^{\prime }-\left (1+3 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.598 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (x +6\right ) y^{\prime }+10 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.51 |
|
|
\[ {}x y^{\prime \prime }+\left (2 x +3\right ) y^{\prime }+8 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.994 |
|
|
\[ {}x \left (1-x \right ) y^{\prime \prime }+2 \left (1-x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Jacobi] |
✓ |
✓ |
1.269 |
|
|
\[ {}x \left (1-x \right ) y^{\prime \prime }+2 \left (1-x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Jacobi] |
✓ |
✓ |
1.359 |
|
|
\[ {}x y^{\prime \prime }+\left (-x +3\right ) y^{\prime }-5 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Laguerre] |
✓ |
✓ |
1.543 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-\left (6 x^{2}-3 x +1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.332 |
|
|
\[ {}x \left (x -2\right )^{2} y^{\prime \prime }-2 \left (x -2\right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.338 |
|
|
\[ {}x \left (x -2\right )^{2} y^{\prime \prime }-2 \left (x -2\right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.986 |
|
|
|
|||||||||
|
\[ {}2 x y^{\prime \prime }+\left (1-x \right ) y^{\prime }-\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.433 |
|
|
\[ {}x y^{\prime \prime }-\left (2+x \right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Laguerre] |
✓ |
✓ |
1.473 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x^{2} y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.102 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x \left (2 x +7\right ) y^{\prime }+2 \left (5+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.256 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+2 x \left (x^{2}+3\right ) y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.035 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-10 x y^{\prime }-18 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer] |
✓ |
✓ |
1.326 |
|
|
\[ {}2 x y^{\prime \prime }+\left (1+2 x \right ) y^{\prime }-3 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.274 |
|
|
\[ {}y^{\prime \prime }+2 x y^{\prime }-8 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_erf] |
✓ |
✓ |
0.324 |
|
|
\[ {}x \left (-x^{2}+1\right ) y^{\prime \prime }-\left (x^{2}+7\right ) y^{\prime }+4 y x = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.088 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x \left (1+2 x \right ) y^{\prime }+\left (1+4 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.275 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-2 x \left (2+x \right ) y^{\prime }+\left (x +3\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.13 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (x^{2}+1\right ) y^{\prime }+\left (-x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.901 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x^{2}-3\right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.766 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x \left (1+2 x \right ) y^{\prime }+\left (1+3 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.741 |
|
|
\[ {}x y^{\prime \prime }+\left (-x^{2}+1\right ) y^{\prime }+2 y x = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.845 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+2 x^{2} y^{\prime }-\left (x +3\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.092 |
|
|
\[ {}x \left (-x^{2}+1\right ) y^{\prime \prime }+5 \left (-x^{2}+1\right ) y^{\prime }-4 y x = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.034 |
|
|
\[ {}x \left (-2 x +1\right ) y^{\prime \prime }-2 \left (2+x \right ) y^{\prime }+18 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.273 |
|
|
\[ {}x y^{\prime }-y+y^{2} = x^{\frac {2}{3}} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
11.943 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x -x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.81 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x -2 x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.452 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x -3 x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.478 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x -x^{2}-x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.525 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x -x^{3}+2 = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.979 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x -x^{4}-6 = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.922 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x -x^{5}+24 = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.902 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x -x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.559 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x -x^{2} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.835 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x -x^{3} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.231 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x -x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.822 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+y x = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Lienard] |
✓ |
✓ |
0.891 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+3 x y^{\prime }-y x = x^{2}+2 x \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.373 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (3 x^{2}+2 x \right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.894 |
|
|
\[ {}2 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+x \left (11 x^{2}+11 x +9\right ) y^{\prime }+\left (7 x^{2}+10 x +6\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.77 |
|
|
\[ {}x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }-x \left (4 x^{2}+3\right ) y^{\prime }+\left (-2 x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.131 |
|
|
\[ {}x y^{\prime \prime }+\left (1+x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.962 |
|
|
\[ {}x^{2} \left (x^{2}-2 x +1\right ) y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+\left (x +4\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.586 |
|
|
\[ {}2 x^{2} \left (2+x \right ) y^{\prime \prime }+5 x^{2} y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.152 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.901 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.898 |
|
|
\[ {}\left (x^{2}-x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.635 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }-\left (x^{2}+\frac {5}{4}\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.873 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.878 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+4 x^{4} y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.858 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }+y^{\prime }+y = x \,{\mathrm e}^{x} \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.553 |
|
|
\[ {}y^{\prime \prime } = \left (x^{2}+3\right ) y \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.548 |
|
|
\[ {}y^{\prime \prime }+2 x y^{\prime }+\left (x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.77 |
|
|
\[ {}y^{\prime \prime }+2 \cot \left (x \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.383 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.622 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+\left (-8 x^{2}+4 x \right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y = 4 \sqrt {x}\, {\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.811 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +2\right ) y^{\prime }+\left (2+x \right ) y = 6 x^{3} {\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.394 |
|
|
\[ {}x y^{\prime \prime }-\left (1+2 x \right ) y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.562 |
|
|
\[ {}y^{\prime \prime }-\frac {y^{\prime }}{\sqrt {x}}+\frac {\left (x +\sqrt {x}-8\right ) y}{4 x^{2}} = x \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.544 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }+y x = x^{1+m} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.669 |
|
|
\[ {}y^{\prime \prime }-\frac {y^{\prime }}{\sqrt {x}}+\frac {\left (x +\sqrt {x}-8\right ) y}{4 x^{2}} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.418 |
|
|
\[ {}\cos \left (x \right )^{2} y^{\prime \prime }-2 \cos \left (x \right ) \sin \left (x \right ) y^{\prime }+\cos \left (x \right )^{2} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[_Lienard] |
✓ |
✓ |
1.16 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-1\right ) y = -3 \,{\mathrm e}^{x^{2}} \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.1 |
|
|
\[ {}y^{\prime \prime }-2 b x y^{\prime }+b^{2} x^{2} y = x \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.674 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-3\right ) y = {\mathrm e}^{x^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.606 |
|
|
\[ {}y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime }+5 y = {\mathrm e}^{x^{2}} \sec \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.567 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+2 \left (x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.708 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x^{5} y^{\prime }+\left (x^{8}+6 x^{4}+4\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.058 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.483 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+y x = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[_Lienard] |
✓ |
✓ |
0.482 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (x +6\right ) y^{\prime }+10 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.856 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.592 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-6 x y^{\prime }+12 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.75 |
|
|
\[ {}\left (x^{2}+3\right ) y^{\prime \prime }-7 x y^{\prime }+16 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.529 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+8 x y^{\prime }+12 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.466 |
|
|
\[ {}3 y^{\prime \prime }+x y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.226 |
|
|
\[ {}5 y^{\prime \prime }-2 x y^{\prime }+10 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.018 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }-3 y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.513 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.353 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.493 |
|
|
\[ {}\left (x^{2}-6 x +10\right ) y^{\prime \prime }-4 \left (x -3\right ) y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.579 |
|
|
\[ {}\left (x^{2}+6 x \right ) y^{\prime \prime }+\left (3 x +9\right ) y^{\prime }-3 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.6 |
|
|
\[ {}t y^{\prime \prime }+\left (t^{2}-1\right ) y^{\prime }+t^{2} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.803 |
|
|
\[ {}t^{2} y^{\prime \prime }-t \left (2+t \right ) y^{\prime }+\left (2+t \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.57 |
|
|
\[ {}t y^{\prime \prime }-\left (1+t \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Laguerre] |
✓ |
✓ |
0.451 |
|
|
\[ {}\left (1-t \right ) y^{\prime \prime }+t y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.47 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.418 |
|
|
\[ {}t y^{\prime \prime }-\left (1+t \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Laguerre] |
✓ |
✓ |
0.347 |
|
|
\[ {}\left (1-t \right ) y^{\prime \prime }+t y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.371 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.342 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.346 |
|
|
\[ {}\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.45 |
|
|
\[ {}2 y^{\prime \prime }+x y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.908 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.336 |
|
|
\[ {}\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.766 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.361 |
|
|
\[ {}\left (-x^{2}+4\right ) y^{\prime \prime }+x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.432 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (-16 x^{2}+3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.419 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.485 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.447 |
|
|
\[ {}\left (x^{2}-2 x \right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }+\left (2 x -2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.809 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+\left (-8 x^{2}+4 x \right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.44 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.307 |
|
|
\[ {}\left (1+2 x \right ) y^{\prime \prime }-2 y^{\prime }-\left (2 x +3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.574 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +2\right ) y^{\prime }+\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.451 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.605 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (-16 x^{2}+3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.347 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}+3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.327 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }-\left (x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.321 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x \left (1+x \right ) y^{\prime }+\left (x^{2}+2 x +2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.403 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x \left (2+x \right ) y^{\prime }+\left (x^{2}+4 x +6\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.365 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (x^{2}+6\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.313 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.507 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x \left (1+x \right ) y^{\prime }+\left (2 x +3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.345 |
|
|
\[ {}\left (3 x -1\right ) y^{\prime \prime }-\left (2+3 x \right ) y^{\prime }-\left (6 x -8\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.57 |
|
|
\[ {}\left (2+x \right ) y^{\prime \prime }+x y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.593 |
|
|
\[ {}x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (x +4\right ) y^{\prime }+\left (2-x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.632 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (1+2 x \right ) y^{\prime }-\left (4+6 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.606 |
|
|
\[ {}x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+x \left (2 x^{2}+4\right ) y^{\prime }+2 \left (-x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.802 |
|
|
\[ {}x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+2 x \left (x^{2}+5\right ) y^{\prime }+2 \left (-x^{2}+3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.863 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+6 x y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.36 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.273 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-8 x y^{\prime }+20 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.303 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-8 x y^{\prime }-12 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.489 |
|
|
\[ {}\left (2 x^{2}+1\right ) y^{\prime \prime }+7 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.908 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-5 x y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.651 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-10 x y^{\prime }+28 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.416 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.613 |
|
|
\[ {}\left (2 x^{2}+1\right ) y^{\prime \prime }-9 x y^{\prime }-6 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.225 |
|
|
\[ {}\left (2 x^{2}-8 x +11\right ) y^{\prime \prime }-16 \left (x -2\right ) y^{\prime }+36 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.291 |
|
|
\[ {}y^{\prime \prime }+\left (x -3\right ) y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.548 |
|
|
\[ {}\left (x^{2}-8 x +14\right ) y^{\prime \prime }-8 \left (x -4\right ) y^{\prime }+20 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.934 |
|
|
\[ {}\left (2 x^{2}+4 x +5\right ) y^{\prime \prime }-20 \left (1+x \right ) y^{\prime }+60 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.176 |
|
|
\[ {}\left (x^{3}+1\right ) y^{\prime \prime }+7 x^{2} y^{\prime }+9 y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.298 |
|
|
\[ {}\left (2 x^{5}+1\right ) y^{\prime \prime }+14 x^{4} y^{\prime }+10 x^{3} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
48.857 |
|
|
\[ {}y^{\prime \prime }+x^{6} y^{\prime }+7 x^{5} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.608 |
|
|
\[ {}\left (x^{8}+1\right ) y^{\prime \prime }-16 x^{7} y^{\prime }+72 x^{6} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
102.757 |
|
|
\[ {}y^{\prime \prime }+x^{5} y^{\prime }+6 x^{4} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.552 |
|
|
\[ {}\left (1+3 x \right ) y^{\prime \prime }+x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
33.077 |
|
|
\[ {}\left (3 x^{2}+x +1\right ) y^{\prime \prime }+\left (2+15 x \right ) y^{\prime }+12 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.211 |
|
|
\[ {}\left (2+x \right ) y^{\prime \prime }+\left (1+x \right ) y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.724 |
|
|
\[ {}\left (x +4\right ) y^{\prime \prime }+\left (2+x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.71 |
|
|
\[ {}\left (2 x^{2}+3 x \right ) y^{\prime \prime }+10 \left (1+x \right ) y^{\prime }+8 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.821 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (6-7 x \right ) y^{\prime }+8 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.321 |
|
|
\[ {}\left (2 x^{2}+x +1\right ) y^{\prime \prime }+\left (1+7 x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.108 |
|
|
\[ {}\left (x +3\right ) y^{\prime \prime }+\left (1+2 x \right ) y^{\prime }-\left (2-x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.522 |
|
|
\[ {}y^{\prime \prime }+3 x y^{\prime }+\left (2 x^{2}+4\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.525 |
|
|
\[ {}\left (4 x +2\right ) y^{\prime \prime }-4 y^{\prime }-\left (4 x +6\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.693 |
|
|
\[ {}y^{\prime \prime }-3 x y^{\prime }+\left (2 x^{2}+5\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.464 |
|
|
\[ {}2 y^{\prime \prime }+5 x y^{\prime }+\left (2 x^{2}+4\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.629 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.237 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.198 |
|
|
\[ {}2 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+x \left (11 x^{2}+11 x +9\right ) y^{\prime }+\left (7 x^{2}+10 x +6\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.757 |
|
|
\[ {}3 x^{2} y^{\prime \prime }+2 x \left (-2 x^{2}+x +1\right ) y^{\prime }+\left (-8 x^{2}+2 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.908 |
|
|
\[ {}12 x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (3 x^{2}+35 x +11\right ) y^{\prime }-\left (-5 x^{2}-10 x +1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.03 |
|
|
\[ {}x^{2} \left (10 x^{2}+x +5\right ) y^{\prime \prime }+x \left (48 x^{2}+3 x +4\right ) y^{\prime }+\left (36 x^{2}+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.125 |
|
|
\[ {}18 x^{2} \left (1+x \right ) y^{\prime \prime }+3 x \left (x^{2}+11 x +5\right ) y^{\prime }-\left (-5 x^{2}-2 x +1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.109 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+x \left (2 x +3\right ) y^{\prime }-\left (1-x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.619 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+x \left (5+x \right ) y^{\prime }-\left (2-3 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.763 |
|
|
\[ {}3 x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.84 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-2 x +1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.642 |
|
|
\[ {}3 x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-\left (1+3 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.44 |
|
|
\[ {}2 x^{2} \left (x +3\right ) y^{\prime \prime }+x \left (1+5 x \right ) y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.94 |
|
|
\[ {}x^{2} \left (x +4\right ) y^{\prime \prime }-x \left (1-3 x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.109 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+5 x y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.533 |
|
|
\[ {}6 x^{2} y^{\prime \prime }+x \left (10-x \right ) y^{\prime }-\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.921 |
|
|
\[ {}x^{2} \left (3+4 x \right ) y^{\prime \prime }+x \left (11+4 x \right ) y^{\prime }-\left (3+4 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.924 |
|
|
\[ {}2 x^{2} \left (2+3 x \right ) y^{\prime \prime }+x \left (4+11 x \right ) y^{\prime }-\left (1-x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.317 |
|
|
\[ {}x^{2} \left (2+x \right ) y^{\prime \prime }+5 x \left (1-x \right ) y^{\prime }-\left (2-8 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.566 |
|
|
\[ {}8 x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+2 x \left (-13 x^{2}+1\right ) y^{\prime }+\left (-9 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.125 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-2 x \left (-x^{2}+2\right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.868 |
|
|
\[ {}x \left (x^{2}+3\right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }-8 y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.889 |
|
|
\[ {}4 x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+x \left (-19 x^{2}+7\right ) y^{\prime }-\left (14 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.618 |
|
|
\[ {}3 x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }+x \left (-11 x^{2}+1\right ) y^{\prime }+\left (-5 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.022 |
|
|
\[ {}2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }-x \left (-7 x^{2}+12\right ) y^{\prime }+\left (3 x^{2}+7\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.871 |
|
|
\[ {}2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+x \left (7 x^{2}+4\right ) y^{\prime }-\left (-3 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.849 |
|
|
\[ {}2 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+5 x \left (6 x^{2}+1\right ) y^{\prime }-\left (-40 x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.262 |
|
|
\[ {}x \left (x^{2}+1\right ) y^{\prime \prime }+\left (7 x^{2}+4\right ) y^{\prime }+8 y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.77 |
|
|
\[ {}2 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (8 x^{2}+3\right ) y^{\prime }-\left (-4 x^{2}+3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.9 |
|
|
\[ {}9 x^{2} y^{\prime \prime }+3 x \left (x^{2}+3\right ) y^{\prime }-\left (-5 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.783 |
|
|
\[ {}6 x^{2} y^{\prime \prime }+x \left (6 x^{2}+1\right ) y^{\prime }+\left (9 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.924 |
|
|
\[ {}9 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+3 x \left (13 x^{2}+3\right ) y^{\prime }-\left (-25 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.891 |
|
|
\[ {}4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+4 x \left (6 x^{2}+1\right ) y^{\prime }-\left (-25 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.843 |
|
|
\[ {}8 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+2 x \left (34 x^{2}+5\right ) y^{\prime }-\left (-30 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.293 |
|
|
\[ {}2 x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (1-3 x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.604 |
|
|
\[ {}6 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+x \left (50 x^{2}+1\right ) y^{\prime }+\left (30 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.524 |
|
|
\[ {}28 x^{2} \left (1-3 x \right ) y^{\prime \prime }-7 x \left (5+9 x \right ) y^{\prime }+7 \left (2+9 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.614 |
|
|
\[ {}8 x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }+2 x \left (-21 x^{2}+10\right ) y^{\prime }-\left (35 x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.524 |
|
|
\[ {}4 x^{2} \left (x^{2}+3 x +1\right ) y^{\prime \prime }-4 x \left (-3 x^{2}-3 x +1\right ) y^{\prime }+3 \left (x^{2}-x +1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.478 |
|
|
\[ {}3 x^{2} \left (1+x \right )^{2} y^{\prime \prime }-x \left (-11 x^{2}-10 x +1\right ) y^{\prime }+\left (5 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.797 |
|
|
\[ {}4 x^{2} \left (x^{2}+2 x +3\right ) y^{\prime \prime }-x \left (-15 x^{2}-14 x +3\right ) y^{\prime }+\left (7 x^{2}+3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.533 |
|
|
\[ {}x^{2} \left (x^{2}-2 x +1\right ) y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+\left (x +4\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.8 |
|
|
\[ {}2 x^{2} \left (2+x \right ) y^{\prime \prime }+5 x^{2} y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.74 |
|
|
\[ {}x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }-2 x \left (2 x^{2}+1\right ) y^{\prime }+\left (-2 x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.913 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (5-x \right ) y^{\prime }+\left (9-4 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.557 |
|
|
\[ {}4 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+12 x^{2} \left (1+x \right ) y^{\prime }+\left (3 x^{2}+3 x +1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
4.439 |
|
|
\[ {}x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }-x \left (-2 x^{2}-4 x +1\right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.125 |
|
|
\[ {}9 x^{2} y^{\prime \prime }+3 x \left (-2 x^{2}+3 x +5\right ) y^{\prime }+\left (-14 x^{2}+12 x +1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.947 |
|
|
\[ {}x^{2} \left (1+2 x \right ) y^{\prime \prime }+x \left (3 x^{2}+14 x +5\right ) y^{\prime }+\left (12 x^{2}+18 x +4\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.084 |
|
|
\[ {}16 x^{2} y^{\prime \prime }+4 x \left (2 x^{2}+x +6\right ) y^{\prime }+\left (18 x^{2}+5 x +1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.953 |
|
|
\[ {}9 x^{2} \left (1+x \right ) y^{\prime \prime }+3 x \left (-x^{2}+11 x +5\right ) y^{\prime }+\left (-7 x^{2}+16 x +1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.063 |
|
|
\[ {}36 x^{2} \left (-2 x +1\right ) y^{\prime \prime }+24 x \left (1-9 x \right ) y^{\prime }+\left (1-70 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.914 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (-x +3\right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.741 |
|
|
\[ {}x^{2} \left (-2 x +1\right ) y^{\prime \prime }-x \left (5-4 x \right ) y^{\prime }+\left (9-4 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.659 |
|
|
\[ {}2 x^{2} \left (2+x \right ) y^{\prime \prime }+x^{2} y^{\prime }+\left (1-x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.841 |
|
|
\[ {}2 x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (6-x \right ) y^{\prime }+\left (8-x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.641 |
|
|
|
|||||||||
|
\[ {}x^{2} \left (1+2 x \right ) y^{\prime \prime }+x \left (5+9 x \right ) y^{\prime }+\left (4+3 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.518 |
|
|
\[ {}x^{2} \left (-2 x +1\right ) y^{\prime \prime }-x \left (5+4 x \right ) y^{\prime }+\left (9+4 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.53 |
|
|
\[ {}x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (7+x \right ) y^{\prime }+\left (9-x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.54 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (-x^{2}+1\right ) y^{\prime }+\left (x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.441 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-3 x \left (-x^{2}+1\right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.652 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+2 x^{3} y^{\prime }+\left (3 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.422 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-2 x^{2}+1\right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.455 |
|
|
\[ {}2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+7 x^{3} y^{\prime }+\left (3 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.181 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-4 x^{2}+1\right ) y^{\prime }+\left (2 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.707 |
|
|
\[ {}4 x^{2} \left (x^{2}+4\right ) y^{\prime \prime }+3 x \left (3 x^{2}+8\right ) y^{\prime }+\left (-9 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.483 |
|
|
\[ {}3 x^{2} \left (x^{2}+3\right ) y^{\prime \prime }+x \left (11 x^{2}+3\right ) y^{\prime }+\left (5 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.032 |
|
|
\[ {}9 x^{2} y^{\prime \prime }-3 x \left (-2 x^{2}+7\right ) y^{\prime }+\left (2 x^{2}+25\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.598 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (-x^{2}+1\right ) y^{\prime }+\left (x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.516 |
|
|
\[ {}x^{2} \left (-2 x +1\right ) y^{\prime \prime }+3 x y^{\prime }+\left (1+4 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.561 |
|
|
\[ {}x \left (1+x \right ) y^{\prime \prime }+\left (1-x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.573 |
|
|
\[ {}x^{2} \left (1-x \right ) y^{\prime \prime }-x \left (3-5 x \right ) y^{\prime }+\left (4-5 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.549 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (9 x^{2}+1\right ) y^{\prime }+\left (25 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.798 |
|
|
\[ {}9 x^{2} y^{\prime \prime }+3 x \left (-x^{2}+1\right ) y^{\prime }+\left (7 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.534 |
|
|
\[ {}x \left (x^{2}+1\right ) y^{\prime \prime }+\left (-x^{2}+1\right ) y^{\prime }-8 y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.753 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+2 x \left (-x^{2}+4\right ) y^{\prime }+\left (7 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.074 |
|
|
\[ {}4 x^{2} \left (1+x \right ) y^{\prime \prime }+8 x^{2} y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.69 |
|
|
\[ {}9 x^{2} \left (x +3\right ) y^{\prime \prime }+3 x \left (3+7 x \right ) y^{\prime }+\left (3+4 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.749 |
|
|
\[ {}x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }-x \left (3 x^{2}+2\right ) y^{\prime }+\left (-x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.379 |
|
|
\[ {}16 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+8 x \left (9 x^{2}+1\right ) y^{\prime }+\left (49 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.709 |
|
|
\[ {}x^{2} \left (4+3 x \right ) y^{\prime \prime }-x \left (4-3 x \right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.36 |
|
|
\[ {}4 x^{2} \left (x^{2}+3 x +1\right ) y^{\prime \prime }+8 x^{2} \left (2 x +3\right ) y^{\prime }+\left (9 x^{2}+3 x +1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.426 |
|
|
\[ {}x^{2} \left (1-x \right )^{2} y^{\prime \prime }-x \left (-3 x^{2}+2 x +1\right ) y^{\prime }+\left (x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.39 |
|
|
\[ {}9 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+3 x \left (13 x^{2}+7 x +1\right ) y^{\prime }+\left (25 x^{2}+4 x +1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.539 |
|
|
\[ {}2 x^{2} \left (2+x \right ) y^{\prime \prime }-x \left (4-7 x \right ) y^{\prime }-\left (5-3 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.695 |
|
|
\[ {}x^{2} \left (-2 x +1\right ) y^{\prime \prime }+x \left (8-9 x \right ) y^{\prime }+\left (6-3 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.689 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (10 x^{2}+3\right ) y^{\prime }-\left (-14 x^{2}+15\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.737 |
|
|
\[ {}x^{2} \left (-2 x^{2}+1\right ) y^{\prime \prime }+x \left (-13 x^{2}+7\right ) y^{\prime }-14 x^{2} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.966 |
|
|
\[ {}4 x^{2} \left (1+x \right ) y^{\prime \prime }+4 x \left (1+2 x \right ) y^{\prime }-\left (1+3 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.529 |
|
|
\[ {}2 x^{2} \left (2+3 x \right ) y^{\prime \prime }+x \left (4+21 x \right ) y^{\prime }-\left (1-9 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.519 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (2+x \right ) y^{\prime }-\left (2-3 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.436 |
|
|
\[ {}4 x^{2} \left (1+x \right ) y^{\prime \prime }+4 x \left (3+8 x \right ) y^{\prime }-\left (5-49 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.648 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (3+10 x \right ) y^{\prime }+30 y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.651 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-3 \left (x +3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.46 |
|
|
\[ {}x^{2} \left (1+2 x \right ) y^{\prime \prime }+x \left (9+13 x \right ) y^{\prime }+\left (7+5 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.642 |
|
|
\[ {}4 x^{2} \left (1+2 x \right ) y^{\prime \prime }-2 x \left (4-x \right ) y^{\prime }-\left (7+5 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.479 |
|
|
\[ {}3 x^{2} \left (x +3\right ) y^{\prime \prime }-x \left (15+x \right ) y^{\prime }-20 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.689 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (1-10 x \right ) y^{\prime }-\left (9-10 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.64 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+3 x^{2} y^{\prime }-\left (6-x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.782 |
|
|
\[ {}x^{2} \left (1+2 x \right ) y^{\prime \prime }-2 x \left (3+14 x \right ) y^{\prime }+\left (6+100 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.515 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (6+11 x \right ) y^{\prime }+\left (6+32 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.619 |
|
|
\[ {}4 x^{2} \left (1+x \right ) y^{\prime \prime }+4 x \left (1+4 x \right ) y^{\prime }-\left (49+27 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.628 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-2 x^{2}+7\right ) y^{\prime }+12 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.664 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (-x^{2}+7\right ) y^{\prime }+12 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.464 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (2 x^{2}+1\right ) y^{\prime }-\left (-10 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.02 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (-2 x^{2}+1\right ) y^{\prime }-4 \left (2 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.532 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (-3 x^{2}+1\right ) y^{\prime }-4 \left (-3 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.018 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (11 x^{2}+5\right ) y^{\prime }+24 x^{2} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.856 |
|
|
\[ {}4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+8 x y^{\prime }-\left (-x^{2}+35\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.782 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-x^{2}+5\right ) y^{\prime }-\left (25 x^{2}+7\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.93 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (2 x^{2}+5\right ) y^{\prime }-21 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.655 |
|
|
\[ {}4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+4 x \left (x^{2}+2\right ) y^{\prime }-\left (x^{2}+15\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.627 |
|
|
\[ {}y^{\prime \prime }-\frac {2 \left (1+t \right ) y^{\prime }}{t^{2}+2 t -1}+\frac {2 y}{t^{2}+2 t -1} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.776 |
|
|
\[ {}y^{\prime \prime }-4 t y^{\prime }+\left (4 t^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.249 |
|
|
\[ {}\left (-t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.464 |
|
|
\[ {}\left (t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.313 |
|
|
\[ {}\left (-t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.45 |
|
|
\[ {}\left (2 t +1\right ) y^{\prime \prime }-4 \left (1+t \right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.493 |
|
|
\[ {}t^{2} y^{\prime \prime }+t y^{\prime }+\left (t^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.36 |
|
|
\[ {}y^{\prime \prime }-\frac {2 t y^{\prime }}{t^{2}+1}+\frac {2 y}{t^{2}+1} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.187 |
|
|
\[ {}y^{\prime \prime }+\left (t^{2}+2 t +1\right ) y^{\prime }-\left (4+4 t \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.143 |
|
|
\[ {}2 t y^{\prime \prime }+\left (1-2 t \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Laguerre] |
✓ |
✓ |
0.572 |
|
|
\[ {}2 t y^{\prime \prime }+\left (1+t \right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.828 |
|
|
\[ {}2 t^{2} y^{\prime \prime }-t y^{\prime }+\left (1+t \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.527 |
|
|
\[ {}2 t^{2} y^{\prime \prime }+\left (t^{2}-t \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.533 |
|
|
\[ {}t^{2} y^{\prime \prime }+\left (-t^{2}+t \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.418 |
|
|
\[ {}t y^{\prime \prime }-\left (t^{2}+2\right ) y^{\prime }+t y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Lienard] |
✓ |
✓ |
0.525 |
|
|
\[ {}t^{2} y^{\prime \prime }+t \left (1+t \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.425 |
|
|
\[ {}t y^{\prime \prime }-\left (4+t \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Laguerre] |
✓ |
✓ |
0.47 |
|
|
\[ {}t^{2} y^{\prime \prime }+\left (t^{2}-3 t \right ) y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.441 |
|
|
\[ {}t y^{\prime \prime }+t y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.439 |
|
|
\[ {}t y^{\prime \prime }+\left (-t^{2}+1\right ) y^{\prime }+4 t y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.413 |
|
|
\[ {}t^{2} y^{\prime \prime }-t \left (1+t \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.43 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+6\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.414 |
|
|
\[ {}\left (-z^{2}+1\right ) y^{\prime \prime }-3 y^{\prime } z +\lambda y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.971 |
|
|
\[ {}4 z y^{\prime \prime }+2 \left (1-z \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.572 |
|
|
\[ {}f^{\prime \prime }+2 \left (z -1\right ) f^{\prime }+4 f = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.435 |
|
|
\[ {}z y^{\prime \prime }-2 y^{\prime }+z y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Lienard] |
✓ |
✓ |
0.53 |
|
|
\[ {}z y^{\prime \prime }+\left (2 z -3\right ) y^{\prime }+\frac {4 y}{z} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.483 |
|
|
\[ {}y^{\prime \prime }+2 x y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_erf] |
✓ |
✓ |
0.326 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.441 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }-3 y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.479 |
|
|
\[ {}\left (-4 x^{2}+1\right ) y^{\prime \prime }-20 x y^{\prime }-16 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.792 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-6 x y^{\prime }+12 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.464 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.812 |
|
|
\[ {}\left (2 x^{2}+1\right ) y^{\prime \prime }+7 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.585 |
|
|
\[ {}4 y^{\prime \prime }+x y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Lienard] |
✓ |
✓ |
0.528 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.219 |
|
|
\[ {}4 x y^{\prime \prime }-x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.267 |
|
|
\[ {}6 x^{2} y^{\prime \prime }+x \left (1+18 x \right ) y^{\prime }+\left (1+12 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.633 |
|
|
\[ {}3 x^{2} y^{\prime \prime }-x \left (x +8\right ) y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
4.878 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x \left (1+2 x \right ) y^{\prime }+2 \left (4 x -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.128 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x^{2} y^{\prime }+\left (1+2 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.432 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (3-2 x \right ) y^{\prime }+\left (-2 x +1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.712 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+\left (4-x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.469 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (-x +3\right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.411 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (2 \sqrt {5}-1\right ) x y^{\prime }+\left (\frac {19}{4}-3 x^{2}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.57 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x -3\right ) y^{\prime }+\left (4-x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.375 |
|
|
\[ {}x^{2} y^{\prime \prime }+x^{2} y^{\prime }-\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.376 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x^{2} y^{\prime }+\left (x -\frac {3}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.399 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+x^{2} y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.461 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x^{2}+6\right ) y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.082 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.423 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.466 |
|
|
\[ {}x^{2} y^{\prime \prime }-x^{2} y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.629 |
|
|
\[ {}x^{2} y^{\prime \prime }-x^{2} y^{\prime }-\left (2+3 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.447 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (5-x \right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.489 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x \left (1-x \right ) y^{\prime }+\left (2 x -9\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.405 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x \left (2+x \right ) y^{\prime }+2 \left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.263 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (1-x \right ) y^{\prime }+\left (1-x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.388 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x \left (1+2 x \right ) y^{\prime }+\left (4 x -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.34 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x +4\right ) y^{\prime }+\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.244 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {9}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.51 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+y x = 0 \] |
1 |
1 |
1 |
kovacic |
[_Lienard] |
✓ |
✓ |
0.328 |
|
|
\[ {}2 x y^{\prime \prime }+5 \left (-2 x +1\right ) y^{\prime }-5 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.112 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.575 |
|
|
\[ {}x y^{\prime \prime }+\left (x +n \right ) y^{\prime }+\left (n +1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.151 |
|
|
\[ {}x^{4} y^{\prime \prime }+x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.652 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (2 x^{2}+x \right ) y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.48 |
|
|
\[ {}\left (4 x^{3}-14 x^{2}-2 x \right ) y^{\prime \prime }-\left (6 x^{2}-7 x +1\right ) y^{\prime }+\left (6 x -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.904 |
|
|
\[ {}x^{2} y^{\prime \prime }+x^{2} y^{\prime }+\left (x -2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.426 |
|
|
\[ {}x^{2} y^{\prime \prime }-x^{2} y^{\prime }+\left (x -2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.395 |
|
|
\[ {}x^{2} \left (1-4 x \right ) y^{\prime \prime }-\frac {x y^{\prime }}{2}-\frac {3 y x}{4} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.023 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }+\left (x -9\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.512 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }+\left (3 x -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.5 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (x^{2}+4 x \right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.466 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-\left (2+3 x \right ) y^{\prime }+\frac {\left (2 x -1\right ) y}{x} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.169 |
|
|
\[ {}x \left (1-x \right ) y^{\prime \prime }+\left (\frac {3}{2}-2 x \right ) y^{\prime }-\frac {y}{4} = 0 \] |
1 |
1 |
1 |
kovacic |
[_Jacobi] |
✓ |
✓ |
0.546 |
|
|
\[ {}2 x \left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.297 |
|
|
\[ {}2 x \left (1-x \right ) y^{\prime \prime }+\left (1-11 x \right ) y^{\prime }-10 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Jacobi] |
✓ |
✓ |
0.555 |
|
|
\[ {}x \left (1-x \right ) y^{\prime \prime }+\frac {\left (-2 x +1\right ) y^{\prime }}{3}+\frac {20 y}{9} = 0 \] |
1 |
1 |
1 |
kovacic |
[_Jacobi] |
✓ |
✓ |
0.563 |
|
|
\[ {}4 y^{\prime \prime }+\frac {3 \left (-x^{2}+2\right ) y}{\left (-x^{2}+1\right )^{2}} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.566 |
|
|
\[ {}u^{\prime \prime }-\frac {2 u^{\prime }}{x}-a^{2} u = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.513 |
|
|
\[ {}u^{\prime \prime }+\frac {2 u^{\prime }}{x}-a^{2} u = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.342 |
|
|
\[ {}u^{\prime \prime }+\frac {2 u^{\prime }}{x}+a^{2} u = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.355 |
|
|
\[ {}u^{\prime \prime }+\frac {4 u^{\prime }}{x}-a^{2} u = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.434 |
|
|
\[ {}u^{\prime \prime }+\frac {4 u^{\prime }}{x}+a^{2} u = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.564 |
|
|
\[ {}y^{\prime \prime }-a^{2} y = \frac {6 y}{x^{2}} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.697 |
|
|
\[ {}y^{\prime \prime }+n^{2} y = \frac {6 y}{x^{2}} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.688 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-\left (x^{2}+\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.32 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\frac {\left (-9 a^{2}+4 x^{2}\right ) y}{4 a^{2}} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.679 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {25}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.527 |
|
|
\[ {}y^{\prime \prime }+q y^{\prime } = \frac {2 y}{x^{2}} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.406 |
|
|
\[ {}x y^{\prime \prime }+3 y^{\prime }+4 x^{3} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.496 |
|
|
\[ {}\left (x^{2}+2 x \right ) y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.457 |
|
|
\[ {}\left (x^{2}+2 x \right ) y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.371 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.319 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.171 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.43 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.188 |
|
|
\[ {}\left (2 x -3\right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.181 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-3 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Hermite] |
✓ |
✓ |
0.441 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.335 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Hermite] |
✓ |
✓ |
0.477 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.316 |
|
|
\[ {}x \left (1+x \right )^{2} y^{\prime \prime }+\left (-x^{2}+1\right ) y^{\prime }+\left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.515 |
|
|
\[ {}2 x y^{\prime \prime }-y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.619 |
|
|
\[ {}x y^{\prime \prime }+x y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.253 |
|
|
\[ {}x \left (-1+x \right )^{2} y^{\prime \prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.447 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+x^{2} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.52 |
|
|
\[ {}x \left (-x^{2}+2\right ) y^{\prime \prime }-\left (x^{2}+4 x +2\right ) \left (\left (1-x \right ) y^{\prime }+y\right ) = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.949 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }-\left (1+2 x \right ) \left (-y+x y^{\prime }\right ) = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.515 |
|
|
\[ {}2 x^{2} \left (2-x \right ) y^{\prime \prime }-x \left (4-x \right ) y^{\prime }+\left (-x +3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.438 |
|
|
\[ {}x^{2} \left (1-x \right ) y^{\prime \prime }+\left (5 x -4\right ) x y^{\prime }+\left (6-9 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.285 |
|
|
\[ {}x y^{\prime \prime }+\left (4 x^{2}+1\right ) y^{\prime }+4 x \left (x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.404 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+8 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.192 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+8 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.169 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+12 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.598 |
|
|
\[ {}x \left (2+x \right ) y^{\prime \prime }+2 \left (1+x \right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.442 |
|
|
\[ {}x \left (2+x \right ) y^{\prime \prime }+\left (1+x \right ) y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.487 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.694 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.366 |
|
|
\[ {}\left (x^{2}-2 x +10\right ) y^{\prime \prime }+x y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.673 |
|
|
\[ {}\left (x^{2}-2 x +10\right ) y^{\prime \prime }+x y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.233 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Hermite] |
✓ |
✓ |
0.876 |
|
|
\[ {}\left (2+x \right ) y^{\prime \prime }+x y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.531 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-6 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.226 |
|
|
\[ {}\left (x^{2}+2\right ) y^{\prime \prime }+3 x y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.7 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.453 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+8 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.167 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (\frac {5}{3} x +x^{2}\right ) y^{\prime }-\frac {y}{3} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.89 |
|
|
\[ {}2 x y^{\prime \prime }-y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.809 |
|
|
\[ {}2 x y^{\prime \prime }-\left (2 x +3\right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Laguerre] |
✓ |
✓ |
1.036 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+3 x y^{\prime }+\left (2 x -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.609 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.336 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.418 |
|
|
\[ {}x y^{\prime \prime }+\left (x -6\right ) y^{\prime }-3 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.512 |
|
|
\[ {}x^{4} y^{\prime \prime }+\lambda y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.321 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}-25\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.733 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (36 x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.401 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.499 |
|
|
\[ {}x y^{\prime \prime }+3 y^{\prime }+x^{3} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.576 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.57 |
|
|
\[ {}16 x^{2} y^{\prime \prime }+32 x y^{\prime }+\left (x^{4}-12\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.601 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }+y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.46 |
|
|
\[ {}x y^{\prime \prime }-\left (2+x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Laguerre] |
✓ |
✓ |
0.375 |
|
|
|
|||||||||
|
\[ {}y^{\prime \prime }+x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.314 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.492 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.228 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+30 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.595 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+y x = 0 \] |
1 |
1 |
1 |
kovacic |
[_Lienard] |
✓ |
✓ |
0.271 |
|
|
\[ {}x y^{\prime \prime }+\left (1+2 x \right ) y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.345 |
|
|
\[ {}2 x \left (-1+x \right ) y^{\prime \prime }-\left (1+x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Jacobi] |
✓ |
✓ |
0.414 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+4 y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.54 |
|
|
\[ {}x y^{\prime \prime }+\left (2-2 x \right ) y^{\prime }+\left (x -2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.389 |
|
|
\[ {}x^{2} y^{\prime \prime }+6 x y^{\prime }+\left (4 x^{2}+6\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.345 |
|
|
\[ {}x y^{\prime \prime }+\left (-2 x +1\right ) y^{\prime }+\left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.372 |
|
|
\[ {}x \left (1-x \right ) y^{\prime \prime }+\left (\frac {1}{2}+2 x \right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Jacobi] |
✓ |
✓ |
0.543 |
|
|
\[ {}4 \left (t^{2}-3 t +2\right ) y^{\prime \prime }-2 y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.655 |
|
|
\[ {}2 \left (t^{2}-5 t +6\right ) y^{\prime \prime }+\left (2 t -3\right ) y^{\prime }-8 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.648 |
|
|
\[ {}3 t \left (1+t \right ) y^{\prime \prime }+t y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.673 |
|
|
\[ {}x^{2} y^{\prime \prime }+\frac {\left (x +\frac {3}{4}\right ) y}{4} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.425 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\frac {\left (x^{2}-1\right ) y}{4} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.365 |
|
|
\[ {}x y^{\prime \prime }+\left (-2 x +1\right ) y^{\prime }+\left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.329 |
|
|
\[ {}x y^{\prime \prime }-\left (1+x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Laguerre] |
✓ |
✓ |
0.615 |
|
|
\[ {}x y^{\prime \prime }+3 y^{\prime }+4 x^{3} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.54 |
|
|
\[ {}x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+2 x \left (-x^{2}+1\right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.314 |
|
|
\[ {}2 x y^{\prime \prime }+\left (x -2\right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.387 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+y x = 0 \] |
1 |
1 |
1 |
kovacic |
[_Lienard] |
✓ |
✓ |
0.317 |
|
|
\[ {}y^{\prime \prime }+2 x^{2} y^{\prime }+\left (x^{4}+2 x -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.36 |
|
|
\[ {}u^{\prime \prime }-\left (1+2 x \right ) u^{\prime }+\left (x^{2}+x -1\right ) u = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.277 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+\left (1+\frac {2}{\left (1+3 x \right )^{2}}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.463 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.305 |
|
|
\[ {}y^{\prime \prime }+\frac {2 y^{\prime }}{x}-\frac {2 y}{\left (1+x \right )^{2}} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.356 |
|
|
\[ {}y^{\prime \prime }+\frac {y}{2 x^{4}} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.473 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.402 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.24 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.242 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.239 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.246 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.244 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.251 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.241 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.242 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.242 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.586 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+y x = 0 \] |
1 |
1 |
1 |
kovacic |
[_Lienard] |
✓ |
✓ |
0.339 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+3 x y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.433 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (3 x^{2}+2 x \right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.415 |
|
|
\[ {}2 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+x \left (11 x^{2}+11 x +9\right ) y^{\prime }+\left (7 x^{2}+10 x +6\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.662 |
|
|
\[ {}x y^{\prime \prime }+\left (1+x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.478 |
|
|
\[ {}x^{2} \left (x^{2}-2 x +1\right ) y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+\left (x +4\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.789 |
|
|
\[ {}2 x^{2} \left (2+x \right ) y^{\prime \prime }+5 x^{2} y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.775 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.362 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.369 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }-\left (x^{2}+\frac {5}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.489 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.572 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+4 x^{4} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.582 |
|
|
\[ {}y^{\prime \prime } = \left (x^{2}+3\right ) y \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.302 |
|
|
\[ {}y^{\prime \prime }+2 x y^{\prime }+\left (x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.227 |
|
|
\[ {}x^{3} y^{\prime \prime }+y^{\prime }-\frac {y}{x} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.246 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.268 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+\left (-8 x^{2}+4 x \right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.352 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.459 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (2+x \right ) y^{\prime }+\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.353 |
|
|
\[ {}\left (1+x \right ) y^{\prime \prime }-\left (2+x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.47 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.669 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.483 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.372 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-6 x y^{\prime }+12 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.442 |
|
|
\[ {}\left (x^{2}+3\right ) y^{\prime \prime }-7 x y^{\prime }+16 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.468 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+8 x y^{\prime }+12 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.429 |
|
|
\[ {}3 y^{\prime \prime }+x y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.164 |
|
|
\[ {}5 y^{\prime \prime }-2 x y^{\prime }+10 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.966 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }-3 y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.468 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.338 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.456 |
|
|
\[ {}\left (x^{2}-6 x +10\right ) y^{\prime \prime }-4 \left (x -3\right ) y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.73 |
|
|
\[ {}\left (x^{2}+6 x \right ) y^{\prime \prime }+\left (3 x +9\right ) y^{\prime }-3 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.516 |
|
|
\[ {}t y^{\prime \prime }+\left (t^{2}-1\right ) y^{\prime }+t^{3} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.745 |
|
|
\[ {}t^{2} y^{\prime \prime }-t \left (2+t \right ) y^{\prime }+\left (2+t \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.314 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.449 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (x -\frac {3}{16}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.361 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.366 |
|
|
\[ {}t^{2} y^{\prime \prime }-t \left (2+t \right ) y^{\prime }+\left (2+t \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.267 |
|
|
\[ {}t y^{\prime \prime }-\left (1+t \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Laguerre] |
✓ |
✓ |
0.369 |
|
|
\[ {}\left (1-t \right ) y^{\prime \prime }+t y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.45 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.276 |
|
|
\[ {}t y^{\prime \prime }-\left (1+t \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Laguerre] |
✓ |
✓ |
0.615 |
|
|
\[ {}\left (1-t \right ) y^{\prime \prime }+t y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.445 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.325 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.334 |
|
|
\[ {}\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.411 |
|
|
\[ {}2 y^{\prime \prime }+x y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.792 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.313 |
|
|
\[ {}\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.463 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.28 |
|
|
\[ {}\left (-x^{2}+4\right ) y^{\prime \prime }+x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.274 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (-16 x^{2}+3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.392 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.694 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.368 |
|
|
\[ {}\left (x^{2}-2 x \right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }+\left (2 x -2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.667 |
|
|
\[ {}\left (1+2 x \right ) y^{\prime \prime }-2 y^{\prime }-\left (2 x +3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.487 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+\left (-8 x^{2}+4 x \right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.345 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.229 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x \left (-1+x \right ) y^{\prime }+\left (x^{2}-2 x +2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.352 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (2 x -1\right ) y^{\prime }+\left (x^{2}-x -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.36 |
|
|
\[ {}\left (-2 x +1\right ) y^{\prime \prime }+2 y^{\prime }+\left (2 x -3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.49 |
|
|
\[ {}2 x y^{\prime \prime }+\left (1+4 x \right ) y^{\prime }+\left (1+2 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.401 |
|
|
\[ {}x y^{\prime \prime }-\left (1+2 x \right ) y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.36 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x \left (1+x \right ) y^{\prime }+\left (2 x +3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.572 |
|
|
\[ {}x y^{\prime \prime }+\left (2-2 x \right ) y^{\prime }+\left (x -2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.359 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +2\right ) y^{\prime }+\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.397 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.37 |
|
|
\[ {}x y^{\prime \prime }-\left (1+4 x \right ) y^{\prime }+\left (4 x +2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.409 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (-16 x^{2}+3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.355 |
|
|
\[ {}\left (1+2 x \right ) x y^{\prime \prime }-2 \left (2 x^{2}-1\right ) y^{\prime }-4 \left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.594 |
|
|
\[ {}\left (x^{2}-2 x \right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }+\left (2 x -2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.701 |
|
|
\[ {}x y^{\prime \prime }-\left (1+4 x \right ) y^{\prime }+\left (4 x +2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.337 |
|
|
\[ {}\left (3 x -1\right ) y^{\prime \prime }-\left (2+3 x \right ) y^{\prime }-\left (6 x -8\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.541 |
|
|
\[ {}\left (1+x \right )^{2} y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }-\left (x^{2}+2 x -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.626 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+\left (-8 x^{2}+4 x \right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.396 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.25 |
|
|
\[ {}\left (1+2 x \right ) y^{\prime \prime }-2 y^{\prime }-\left (2 x +3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.55 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +2\right ) y^{\prime }+\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.379 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.351 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (-16 x^{2}+3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.332 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}+3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.317 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }-\left (x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.324 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x \left (1+x \right ) y^{\prime }+\left (x^{2}+2 x +2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.352 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x \left (2+x \right ) y^{\prime }+\left (x^{2}+4 x +6\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.374 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (x^{2}+6\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.635 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.513 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x \left (1+x \right ) y^{\prime }+\left (2 x +3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.325 |
|
|
\[ {}\left (3 x -1\right ) y^{\prime \prime }-\left (2+3 x \right ) y^{\prime }-\left (6 x -8\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.53 |
|
|
\[ {}\left (2+x \right ) y^{\prime \prime }+x y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.543 |
|
|
\[ {}x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (x +4\right ) y^{\prime }+\left (2-x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.326 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (1+2 x \right ) y^{\prime }-\left (4+6 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.613 |
|
|
\[ {}x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+x \left (2 x^{2}+4\right ) y^{\prime }+2 \left (-x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.737 |
|
|
\[ {}x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+2 x \left (x^{2}+5\right ) y^{\prime }+2 \left (-x^{2}+3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.803 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+6 x y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.339 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.248 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-8 x y^{\prime }+20 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.556 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-8 x y^{\prime }-12 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.488 |
|
|
\[ {}\left (2 x^{2}+1\right ) y^{\prime \prime }+7 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.851 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-5 x y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.592 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-10 x y^{\prime }+28 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.398 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.385 |
|
|
\[ {}\left (2 x^{2}-8 x +11\right ) y^{\prime \prime }-16 \left (x -2\right ) y^{\prime }+36 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.174 |
|
|
\[ {}y^{\prime \prime }+\left (x -3\right ) y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.493 |
|
|
\[ {}\left (x^{2}-8 x +14\right ) y^{\prime \prime }-8 \left (x -4\right ) y^{\prime }+20 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.891 |
|
|
\[ {}\left (2 x^{2}+4 x +5\right ) y^{\prime \prime }-20 \left (1+x \right ) y^{\prime }+60 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.085 |
|
|
\[ {}\left (x^{3}+1\right ) y^{\prime \prime }+7 x^{2} y^{\prime }+9 y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.181 |
|
|
\[ {}\left (2 x^{5}+1\right ) y^{\prime \prime }+14 x^{4} y^{\prime }+10 x^{3} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
40.151 |
|
|
\[ {}y^{\prime \prime }+x^{6} y^{\prime }+7 x^{5} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.716 |
|
|
\[ {}\left (x^{8}+1\right ) y^{\prime \prime }-16 x^{7} y^{\prime }+72 x^{6} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
98.624 |
|
|
\[ {}y^{\prime \prime }+x^{5} y^{\prime }+6 x^{4} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.551 |
|
|
\[ {}\left (1+3 x \right ) y^{\prime \prime }+x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
33.325 |
|
|
\[ {}\left (3 x^{2}+x +1\right ) y^{\prime \prime }+\left (2+15 x \right ) y^{\prime }+12 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.952 |
|
|
\[ {}\left (2+x \right ) y^{\prime \prime }+\left (1+x \right ) y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.564 |
|
|
\[ {}\left (x +4\right ) y^{\prime \prime }+\left (2+x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.569 |
|
|
\[ {}\left (2 x^{2}+3 x \right ) y^{\prime \prime }+10 \left (1+x \right ) y^{\prime }+8 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.637 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (6-7 x \right ) y^{\prime }+8 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.323 |
|
|
\[ {}\left (2 x^{2}+x +1\right ) y^{\prime \prime }+\left (1+7 x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.78 |
|
|
\[ {}\left (x +3\right ) y^{\prime \prime }+\left (1+2 x \right ) y^{\prime }-\left (2-x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.691 |
|
|
\[ {}y^{\prime \prime }+3 x y^{\prime }+\left (2 x^{2}+4\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.869 |
|
|
\[ {}\left (4 x +2\right ) y^{\prime \prime }-4 y^{\prime }-\left (4 x +6\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.56 |
|
|
\[ {}y^{\prime \prime }-3 x y^{\prime }+\left (2 x^{2}+5\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.47 |
|
|
\[ {}2 y^{\prime \prime }+5 x y^{\prime }+\left (2 x^{2}+4\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.426 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.281 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.227 |
|
|
\[ {}2 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+x \left (11 x^{2}+11 x +9\right ) y^{\prime }+\left (7 x^{2}+10 x +6\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.554 |
|
|
\[ {}3 x^{2} y^{\prime \prime }+2 x \left (-2 x^{2}+x +1\right ) y^{\prime }+\left (-8 x^{2}+2 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.825 |
|
|
\[ {}12 x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (3 x^{2}+35 x +11\right ) y^{\prime }-\left (-5 x^{2}-10 x +1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.974 |
|
|
\[ {}18 x^{2} \left (1+x \right ) y^{\prime \prime }+3 x \left (x^{2}+11 x +5\right ) y^{\prime }-\left (-5 x^{2}-2 x +1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.444 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+x \left (2 x +3\right ) y^{\prime }-\left (1-x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.613 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+x \left (5+x \right ) y^{\prime }-\left (2-3 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.62 |
|
|
\[ {}3 x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.821 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-2 x +1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.472 |
|
|
\[ {}3 x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-\left (1+3 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.912 |
|
|
\[ {}2 x^{2} \left (x +3\right ) y^{\prime \prime }+x \left (1+5 x \right ) y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.935 |
|
|
\[ {}x^{2} \left (x +4\right ) y^{\prime \prime }-x \left (1-3 x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.093 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+5 x y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.513 |
|
|
\[ {}6 x^{2} y^{\prime \prime }+x \left (10-x \right ) y^{\prime }-\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.866 |
|
|
\[ {}x^{2} \left (3+4 x \right ) y^{\prime \prime }+x \left (11+4 x \right ) y^{\prime }-\left (3+4 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.867 |
|
|
\[ {}2 x^{2} \left (2+3 x \right ) y^{\prime \prime }+x \left (4+11 x \right ) y^{\prime }-\left (1-x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.353 |
|
|
\[ {}x^{2} \left (2+x \right ) y^{\prime \prime }+5 x \left (1-x \right ) y^{\prime }-\left (2-8 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.407 |
|
|
\[ {}8 x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+2 x \left (-13 x^{2}+1\right ) y^{\prime }+\left (-9 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.336 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-2 x \left (-x^{2}+2\right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.831 |
|
|
\[ {}x \left (x^{2}+3\right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }-8 y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.818 |
|
|
\[ {}4 x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+x \left (-19 x^{2}+7\right ) y^{\prime }-\left (14 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.339 |
|
|
\[ {}3 x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }+x \left (-11 x^{2}+1\right ) y^{\prime }+\left (-5 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.046 |
|
|
\[ {}2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }-x \left (-7 x^{2}+12\right ) y^{\prime }+\left (3 x^{2}+7\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.888 |
|
|
\[ {}2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+x \left (7 x^{2}+4\right ) y^{\prime }-\left (-3 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.856 |
|
|
\[ {}2 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+5 x \left (6 x^{2}+1\right ) y^{\prime }-\left (-40 x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.277 |
|
|
\[ {}x \left (x^{2}+1\right ) y^{\prime \prime }+\left (7 x^{2}+4\right ) y^{\prime }+8 y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.747 |
|
|
\[ {}2 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (8 x^{2}+3\right ) y^{\prime }-\left (-4 x^{2}+3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.323 |
|
|
\[ {}9 x^{2} y^{\prime \prime }+3 x \left (x^{2}+3\right ) y^{\prime }-\left (-5 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.8 |
|
|
\[ {}6 x^{2} y^{\prime \prime }+x \left (6 x^{2}+1\right ) y^{\prime }+\left (9 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.919 |
|
|
\[ {}9 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+3 x \left (13 x^{2}+3\right ) y^{\prime }-\left (-25 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.868 |
|
|
\[ {}4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+4 x \left (6 x^{2}+1\right ) y^{\prime }-\left (-25 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.74 |
|
|
\[ {}8 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+2 x \left (34 x^{2}+5\right ) y^{\prime }-\left (-30 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.029 |
|
|
\[ {}2 x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (1-3 x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.7 |
|
|
\[ {}6 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+x \left (50 x^{2}+1\right ) y^{\prime }+\left (30 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.52 |
|
|
\[ {}28 x^{2} \left (1-3 x \right ) y^{\prime \prime }-7 x \left (5+9 x \right ) y^{\prime }+7 \left (2+9 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.596 |
|
|
\[ {}8 x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }+2 x \left (-21 x^{2}+10\right ) y^{\prime }-\left (35 x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.545 |
|
|
|
|||||||||
|
\[ {}4 x^{2} \left (x^{2}+3 x +1\right ) y^{\prime \prime }-4 x \left (-3 x^{2}-3 x +1\right ) y^{\prime }+3 \left (x^{2}-x +1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.491 |
|
|
\[ {}3 x^{2} \left (1+x \right )^{2} y^{\prime \prime }-x \left (-11 x^{2}-10 x +1\right ) y^{\prime }+\left (5 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.871 |
|
|
\[ {}4 x^{2} \left (x^{2}+2 x +3\right ) y^{\prime \prime }-x \left (-15 x^{2}-14 x +3\right ) y^{\prime }+\left (7 x^{2}+3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.482 |
|
|
\[ {}x^{2} \left (x^{2}-2 x +1\right ) y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+\left (x +4\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.691 |
|
|
\[ {}2 x^{2} \left (2+x \right ) y^{\prime \prime }+5 x^{2} y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.643 |
|
|
\[ {}x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }-2 x \left (2 x^{2}+1\right ) y^{\prime }+\left (-2 x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.7 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (5-x \right ) y^{\prime }+\left (9-4 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.546 |
|
|
\[ {}4 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+12 x^{2} \left (1+x \right ) y^{\prime }+\left (3 x^{2}+3 x +1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
4.57 |
|
|
\[ {}x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }-x \left (-2 x^{2}-4 x +1\right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.25 |
|
|
\[ {}9 x^{2} y^{\prime \prime }+3 x \left (-2 x^{2}+3 x +5\right ) y^{\prime }+\left (-14 x^{2}+12 x +1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.977 |
|
|
\[ {}x^{2} \left (1+2 x \right ) y^{\prime \prime }+x \left (3 x^{2}+14 x +5\right ) y^{\prime }+\left (12 x^{2}+18 x +4\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.122 |
|
|
\[ {}16 x^{2} y^{\prime \prime }+4 x \left (2 x^{2}+x +6\right ) y^{\prime }+\left (18 x^{2}+5 x +1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.98 |
|
|
\[ {}9 x^{2} \left (1+x \right ) y^{\prime \prime }+3 x \left (-x^{2}+11 x +5\right ) y^{\prime }+\left (-7 x^{2}+16 x +1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.548 |
|
|
\[ {}36 x^{2} \left (-2 x +1\right ) y^{\prime \prime }+24 x \left (1-9 x \right ) y^{\prime }+\left (1-70 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.913 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (-x +3\right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.74 |
|
|
\[ {}x^{2} \left (-2 x +1\right ) y^{\prime \prime }-x \left (5-4 x \right ) y^{\prime }+\left (9-4 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.532 |
|
|
\[ {}2 x^{2} \left (2+x \right ) y^{\prime \prime }+x^{2} y^{\prime }+\left (1-x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.746 |
|
|
\[ {}2 x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (6-x \right ) y^{\prime }+\left (8-x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.74 |
|
|
\[ {}x^{2} \left (1+2 x \right ) y^{\prime \prime }+x \left (5+9 x \right ) y^{\prime }+\left (4+3 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.695 |
|
|
\[ {}x^{2} \left (-2 x +1\right ) y^{\prime \prime }-x \left (5+4 x \right ) y^{\prime }+\left (9+4 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.699 |
|
|
\[ {}x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (7+x \right ) y^{\prime }+\left (9-x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.701 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (-x^{2}+1\right ) y^{\prime }+\left (x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.485 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-3 x \left (-x^{2}+1\right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.76 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+2 x^{3} y^{\prime }+\left (3 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.679 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-2 x^{2}+1\right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.502 |
|
|
\[ {}2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+7 x^{3} y^{\prime }+\left (3 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.158 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-4 x^{2}+1\right ) y^{\prime }+\left (2 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.684 |
|
|
\[ {}4 x^{2} \left (x^{2}+4\right ) y^{\prime \prime }+3 x \left (3 x^{2}+8\right ) y^{\prime }+\left (-9 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.963 |
|
|
\[ {}3 x^{2} \left (x^{2}+3\right ) y^{\prime \prime }+x \left (11 x^{2}+3\right ) y^{\prime }+\left (5 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.001 |
|
|
\[ {}9 x^{2} y^{\prime \prime }-3 x \left (-2 x^{2}+7\right ) y^{\prime }+\left (2 x^{2}+25\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.563 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (-x^{2}+1\right ) y^{\prime }+\left (x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.464 |
|
|
\[ {}x^{2} \left (-2 x +1\right ) y^{\prime \prime }+3 x y^{\prime }+\left (1+4 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.548 |
|
|
\[ {}x \left (1+x \right ) y^{\prime \prime }+\left (1-x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.557 |
|
|
\[ {}x^{2} \left (1-x \right ) y^{\prime \prime }-x \left (3-5 x \right ) y^{\prime }+\left (4-5 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.545 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (9 x^{2}+1\right ) y^{\prime }+\left (25 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.836 |
|
|
\[ {}9 x^{2} y^{\prime \prime }+3 x \left (-x^{2}+1\right ) y^{\prime }+\left (7 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.796 |
|
|
\[ {}x \left (x^{2}+1\right ) y^{\prime \prime }+\left (-x^{2}+1\right ) y^{\prime }-8 y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.802 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+2 x \left (-x^{2}+4\right ) y^{\prime }+\left (7 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.037 |
|
|
\[ {}4 x^{2} \left (1+x \right ) y^{\prime \prime }+8 x^{2} y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.538 |
|
|
\[ {}9 x^{2} \left (x +3\right ) y^{\prime \prime }+3 x \left (3+7 x \right ) y^{\prime }+\left (3+4 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.851 |
|
|
\[ {}x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }-x \left (3 x^{2}+2\right ) y^{\prime }+\left (-x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.431 |
|
|
\[ {}16 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+8 x \left (9 x^{2}+1\right ) y^{\prime }+\left (49 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.813 |
|
|
\[ {}x^{2} \left (4+3 x \right ) y^{\prime \prime }-x \left (4-3 x \right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.414 |
|
|
\[ {}4 x^{2} \left (x^{2}+3 x +1\right ) y^{\prime \prime }+8 x^{2} \left (2 x +3\right ) y^{\prime }+\left (9 x^{2}+3 x +1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.49 |
|
|
\[ {}x^{2} \left (1-x \right )^{2} y^{\prime \prime }-x \left (-3 x^{2}+2 x +1\right ) y^{\prime }+\left (x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.454 |
|
|
\[ {}9 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+3 x \left (13 x^{2}+7 x +1\right ) y^{\prime }+\left (25 x^{2}+4 x +1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.816 |
|
|
\[ {}2 x^{2} \left (2+x \right ) y^{\prime \prime }-x \left (4-7 x \right ) y^{\prime }-\left (5-3 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.713 |
|
|
\[ {}x^{2} \left (-2 x +1\right ) y^{\prime \prime }+x \left (8-9 x \right ) y^{\prime }+\left (6-3 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.681 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (10 x^{2}+3\right ) y^{\prime }-\left (-14 x^{2}+15\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.743 |
|
|
\[ {}x^{2} \left (-2 x^{2}+1\right ) y^{\prime \prime }+x \left (-13 x^{2}+7\right ) y^{\prime }-14 x^{2} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.804 |
|
|
\[ {}4 x^{2} \left (1+x \right ) y^{\prime \prime }+4 x \left (1+2 x \right ) y^{\prime }-\left (1+3 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.495 |
|
|
\[ {}2 x^{2} \left (2+3 x \right ) y^{\prime \prime }+x \left (4+21 x \right ) y^{\prime }-\left (1-9 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.516 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (2+x \right ) y^{\prime }-\left (2-3 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.431 |
|
|
\[ {}4 x^{2} \left (1+x \right ) y^{\prime \prime }+4 x \left (3+8 x \right ) y^{\prime }-\left (5-49 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.652 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (3+10 x \right ) y^{\prime }+30 y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.633 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-3 \left (x +3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.47 |
|
|
\[ {}x^{2} \left (1+2 x \right ) y^{\prime \prime }+x \left (9+13 x \right ) y^{\prime }+\left (7+5 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.789 |
|
|
\[ {}4 x^{2} \left (1+2 x \right ) y^{\prime \prime }-2 x \left (4-x \right ) y^{\prime }-\left (7+5 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.509 |
|
|
\[ {}3 x^{2} \left (x +3\right ) y^{\prime \prime }-x \left (15+x \right ) y^{\prime }-20 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.688 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (1-10 x \right ) y^{\prime }-\left (9-10 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.632 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+3 x^{2} y^{\prime }-\left (6-x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.508 |
|
|
\[ {}x^{2} \left (1+2 x \right ) y^{\prime \prime }-2 x \left (3+14 x \right ) y^{\prime }+\left (6+100 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.471 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (6+11 x \right ) y^{\prime }+\left (6+32 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.609 |
|
|
\[ {}4 x^{2} \left (1+x \right ) y^{\prime \prime }+4 x \left (1+4 x \right ) y^{\prime }-\left (49+27 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.629 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-2 x^{2}+7\right ) y^{\prime }+12 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.666 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (-x^{2}+7\right ) y^{\prime }+12 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.455 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (2 x^{2}+1\right ) y^{\prime }-\left (-10 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.009 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (-2 x^{2}+1\right ) y^{\prime }-4 \left (2 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.724 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (-3 x^{2}+1\right ) y^{\prime }-4 \left (-3 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.325 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (11 x^{2}+5\right ) y^{\prime }+24 x^{2} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.875 |
|
|
\[ {}4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+8 x y^{\prime }-\left (-x^{2}+35\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.78 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-x^{2}+5\right ) y^{\prime }-\left (25 x^{2}+7\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.761 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (2 x^{2}+5\right ) y^{\prime }-21 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.771 |
|
|
\[ {}4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+4 x \left (x^{2}+2\right ) y^{\prime }-\left (x^{2}+15\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.735 |
|
|
\[ {}y^{\prime \prime }-\frac {2 \left (1+t \right ) y^{\prime }}{t^{2}+2 t -1}+\frac {2 y}{t^{2}+2 t -1} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.881 |
|
|
\[ {}y^{\prime \prime }-4 t y^{\prime }+\left (4 t^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.284 |
|
|
\[ {}\left (-t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.54 |
|
|
\[ {}\left (t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.344 |
|
|
\[ {}\left (-t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.743 |
|
|
\[ {}\left (2 t +1\right ) y^{\prime \prime }-4 \left (1+t \right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.483 |
|
|
\[ {}t^{2} y^{\prime \prime }+t y^{\prime }+\left (t^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.359 |
|
|
\[ {}y^{\prime \prime }-\frac {2 t y^{\prime }}{t^{2}+1}+\frac {2 y}{t^{2}+1} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.349 |
|
|
\[ {}y^{\prime \prime }+\left (t^{2}+2 t +1\right ) y^{\prime }-\left (4+4 t \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.99 |
|
|
\[ {}2 t y^{\prime \prime }+\left (1-2 t \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Laguerre] |
✓ |
✓ |
0.564 |
|
|
\[ {}2 t y^{\prime \prime }+\left (1+t \right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.841 |
|
|
\[ {}2 t^{2} y^{\prime \prime }-t y^{\prime }+\left (1+t \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.546 |
|
|
\[ {}2 t^{2} y^{\prime \prime }+\left (t^{2}-t \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.534 |
|
|
\[ {}t^{2} y^{\prime \prime }+\left (-t^{2}+t \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.425 |
|
|
\[ {}t y^{\prime \prime }-\left (t^{2}+2\right ) y^{\prime }+t y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Lienard] |
✓ |
✓ |
0.553 |
|
|
\[ {}t^{2} y^{\prime \prime }+t \left (1+t \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.602 |
|
|
\[ {}t y^{\prime \prime }-\left (4+t \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Laguerre] |
✓ |
✓ |
0.444 |
|
|
\[ {}t^{2} y^{\prime \prime }+\left (t^{2}-3 t \right ) y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.388 |
|
|
\[ {}t y^{\prime \prime }+t y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.382 |
|
|
\[ {}t y^{\prime \prime }+\left (-t^{2}+1\right ) y^{\prime }+4 t y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.221 |
|
|
\[ {}t^{2} y^{\prime \prime }-t \left (1+t \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.416 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+6\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.421 |
|
|
\[ {}\left (-z^{2}+1\right ) y^{\prime \prime }-3 y^{\prime } z +y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
1.111 |
|
|
\[ {}4 z y^{\prime \prime }+2 \left (1-z \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.592 |
|
|
\[ {}f^{\prime \prime }+2 \left (z -1\right ) f^{\prime }+4 f = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.449 |
|
|
\[ {}z y^{\prime \prime }-2 y^{\prime }+z y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Lienard] |
✓ |
✓ |
0.561 |
|
|
\[ {}z y^{\prime \prime }+\left (2 z -3\right ) y^{\prime }+\frac {4 y}{z} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.741 |
|
|
\[ {}x y^{\prime \prime }+\left (-2 x +1\right ) y^{\prime }+\left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.376 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.369 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.499 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}-1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.322 |
|
|
\[ {}x y^{\prime \prime }-\left (1+2 x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.391 |
|
|
\[ {}y^{\prime \prime }+2 x y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_erf] |
✓ |
✓ |
0.325 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.858 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }-3 y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.513 |
|
|
\[ {}\left (-4 x^{2}+1\right ) y^{\prime \prime }-20 x y^{\prime }-16 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.76 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-6 x y^{\prime }+12 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.561 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.211 |
|
|
\[ {}\left (2 x^{2}+1\right ) y^{\prime \prime }+7 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.63 |
|
|
\[ {}4 y^{\prime \prime }+x y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Lienard] |
✓ |
✓ |
0.526 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.165 |
|
|
\[ {}4 x y^{\prime \prime }-x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.277 |
|
|
\[ {}6 x^{2} y^{\prime \prime }+x \left (1+18 x \right ) y^{\prime }+\left (1+12 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.643 |
|
|
\[ {}3 x^{2} y^{\prime \prime }-x \left (x +8\right ) y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
4.008 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x \left (1+2 x \right ) y^{\prime }+2 \left (4 x -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.101 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x^{2} y^{\prime }+\left (1+2 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.424 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (3-2 x \right ) y^{\prime }+\left (-2 x +1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.488 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+\left (4-x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.487 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (-x +3\right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.691 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (2 \sqrt {5}-1\right ) x y^{\prime }+\left (\frac {19}{4}-3 x^{2}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.572 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x -3\right ) y^{\prime }+\left (4-x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.378 |
|
|
\[ {}x^{2} y^{\prime \prime }+x^{2} y^{\prime }-\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.364 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x^{2} y^{\prime }+\left (x -\frac {3}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.403 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+x^{2} y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.476 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x^{2}+6\right ) y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.059 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.421 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.47 |
|
|
\[ {}x^{2} y^{\prime \prime }-x^{2} y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.421 |
|
|
\[ {}x^{2} y^{\prime \prime }-x^{2} y^{\prime }-\left (2+3 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.49 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (5-x \right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.755 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x \left (1-x \right ) y^{\prime }+\left (2 x -9\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.428 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x \left (2+x \right ) y^{\prime }+2 \left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.28 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (1-x \right ) y^{\prime }+\left (1-x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.387 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x \left (1+2 x \right ) y^{\prime }+\left (4 x -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.332 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x +4\right ) y^{\prime }+\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.237 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {9}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.504 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+y x = 0 \] |
1 |
1 |
1 |
kovacic |
[_Lienard] |
✓ |
✓ |
0.319 |
|
|
\[ {}2 x y^{\prime \prime }+5 \left (-2 x +1\right ) y^{\prime }-5 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.085 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.361 |
|
|
\[ {}x y^{\prime \prime }+\left (x +n \right ) y^{\prime }+\left (n +1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.793 |
|
|
\[ {}x^{4} y^{\prime \prime }+x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.151 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (2 x^{2}+x \right ) y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.441 |
|
|
\[ {}\left (4 x^{3}-14 x^{2}-2 x \right ) y^{\prime \prime }-\left (6 x^{2}-7 x +1\right ) y^{\prime }+\left (6 x -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.845 |
|
|
\[ {}x^{2} y^{\prime \prime }+x^{2} y^{\prime }+\left (x -2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.39 |
|
|
\[ {}x^{2} y^{\prime \prime }-x^{2} y^{\prime }+\left (x -2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.36 |
|
|
\[ {}x^{2} \left (1-4 x \right ) y^{\prime \prime }+\left (-\frac {1}{4} x -x^{2}\right ) y^{\prime }-\frac {5 y x}{16} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.045 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }+\left (x -9\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.501 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }+\left (3 x -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.477 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (x^{2}+4 x \right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.461 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-\left (2+3 x \right ) y^{\prime }+\frac {\left (2 x -1\right ) y}{x} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.602 |
|
|
\[ {}x \left (1-x \right ) y^{\prime \prime }+\left (\frac {3}{2}-2 x \right ) y^{\prime }-\frac {y}{4} = 0 \] |
1 |
1 |
1 |
kovacic |
[_Jacobi] |
✓ |
✓ |
0.541 |
|
|
\[ {}2 x \left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.52 |
|
|
\[ {}2 x \left (1-x \right ) y^{\prime \prime }+\left (1-11 x \right ) y^{\prime }-10 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Jacobi] |
✓ |
✓ |
0.503 |
|
|
\[ {}x \left (1-x \right ) y^{\prime \prime }+\frac {\left (-2 x +1\right ) y^{\prime }}{3}+\frac {20 y}{9} = 0 \] |
1 |
1 |
1 |
kovacic |
[_Jacobi] |
✓ |
✓ |
0.51 |
|
|
\[ {}4 y^{\prime \prime }+\frac {3 \left (-x^{2}+2\right ) y}{\left (-x^{2}+1\right )^{2}} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.506 |
|
|
\[ {}u^{\prime \prime }-\frac {2 u^{\prime }}{x}-a^{2} u = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.461 |
|
|
\[ {}u^{\prime \prime }+\frac {2 u^{\prime }}{x}-a^{2} u = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.296 |
|
|
\[ {}u^{\prime \prime }+\frac {2 u^{\prime }}{x}+a^{2} u = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.296 |
|
|
\[ {}u^{\prime \prime }+\frac {4 u^{\prime }}{x}-a^{2} u = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.391 |
|
|
\[ {}u^{\prime \prime }+\frac {4 u^{\prime }}{x}+a^{2} u = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.491 |
|
|
\[ {}y^{\prime \prime }-a^{2} y = \frac {6 y}{x^{2}} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.428 |
|
|
\[ {}y^{\prime \prime }+n^{2} y = \frac {6 y}{x^{2}} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.697 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-\left (x^{2}+\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.562 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\frac {\left (-9 a^{2}+4 x^{2}\right ) y}{4 a^{2}} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.773 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {25}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.531 |
|
|
\[ {}y^{\prime \prime }+q y^{\prime } = \frac {2 y}{x^{2}} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.421 |
|
|
\[ {}x y^{\prime \prime }+3 y^{\prime }+4 x^{3} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.492 |
|
|
\[ {}x^{2} \left (2-x \right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.438 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.326 |
|
|
\[ {}x y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.362 |
|
|
\[ {}3 x y^{\prime \prime }-2 \left (3 x -1\right ) y^{\prime }+\left (-2+3 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.47 |
|
|
\[ {}x \left (1+x \right ) y^{\prime \prime }-\left (-1+x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.47 |
|
|
\[ {}\left (x^{2}+2 x \right ) y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.451 |
|
|
\[ {}\left (x^{2}+2 x \right ) y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.688 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.342 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.174 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.253 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.187 |
|
|
\[ {}\left (2 x -3\right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.117 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-3 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Hermite] |
✓ |
✓ |
0.433 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.305 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Hermite] |
✓ |
✓ |
0.487 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.128 |
|
|
\[ {}x \left (1+x \right )^{2} y^{\prime \prime }+\left (-x^{2}+1\right ) y^{\prime }+\left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.511 |
|
|
\[ {}2 x y^{\prime \prime }-y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.792 |
|
|
\[ {}x y^{\prime \prime }+x y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.249 |
|
|
\[ {}x \left (-1+x \right )^{2} y^{\prime \prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.408 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+x^{2} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.324 |
|
|
\[ {}x \left (-x^{2}+2\right ) y^{\prime \prime }-\left (x^{2}+4 x +2\right ) \left (\left (1-x \right ) y^{\prime }+y\right ) = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.93 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }-\left (1+2 x \right ) \left (-y+x y^{\prime }\right ) = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.531 |
|
|
\[ {}2 x^{2} \left (2-x \right ) y^{\prime \prime }-x \left (4-x \right ) y^{\prime }+\left (-x +3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.473 |
|
|
\[ {}x^{2} \left (1-x \right ) y^{\prime \prime }+\left (5 x -4\right ) x y^{\prime }+\left (6-9 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.303 |
|
|
\[ {}x y^{\prime \prime }+\left (4 x^{2}+1\right ) y^{\prime }+4 x \left (x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.408 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+8 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.166 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+8 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.162 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+12 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.824 |
|
|
|
|||||||||
|
\[ {}x \left (2+x \right ) y^{\prime \prime }+2 \left (1+x \right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.495 |
|
|
\[ {}x \left (2+x \right ) y^{\prime \prime }+\left (1+x \right ) y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.492 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.453 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.352 |
|
|
\[ {}\left (x^{2}-2 x +10\right ) y^{\prime \prime }+x y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.7 |
|
|
\[ {}\left (x^{2}-2 x +10\right ) y^{\prime \prime }+x y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.252 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Hermite] |
✓ |
✓ |
0.948 |
|
|
\[ {}\left (2+x \right ) y^{\prime \prime }+x y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.548 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-6 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.233 |
|
|
\[ {}\left (x^{2}+2\right ) y^{\prime \prime }+3 x y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.696 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.729 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+8 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.162 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (\frac {5}{3} x +x^{2}\right ) y^{\prime }-\frac {y}{3} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.194 |
|
|
\[ {}2 x y^{\prime \prime }-y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.653 |
|
|
\[ {}2 x y^{\prime \prime }-\left (2 x +3\right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Laguerre] |
✓ |
✓ |
0.753 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+3 x y^{\prime }+\left (2 x -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.643 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.339 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.457 |
|
|
\[ {}x y^{\prime \prime }+\left (x -6\right ) y^{\prime }-3 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.532 |
|
|
\[ {}x^{4} y^{\prime \prime }+\lambda y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.334 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}-25\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.786 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (36 x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.661 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.532 |
|
|
\[ {}x y^{\prime \prime }+3 y^{\prime }+x^{3} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.534 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.329 |
|
|
\[ {}16 x^{2} y^{\prime \prime }+32 x y^{\prime }+\left (x^{4}-12\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.518 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }+y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.474 |
|
|
\[ {}x y^{\prime \prime }-\left (2+x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Laguerre] |
✓ |
✓ |
0.4 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.33 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.511 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.245 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+30 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.625 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+y x = 0 \] |
1 |
1 |
1 |
kovacic |
[_Lienard] |
✓ |
✓ |
0.545 |
|
|
\[ {}x y^{\prime \prime }+\left (1+2 x \right ) y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.445 |
|
|
\[ {}2 x \left (-1+x \right ) y^{\prime \prime }-\left (1+x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Jacobi] |
✓ |
✓ |
0.474 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+4 y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.316 |
|
|
\[ {}x y^{\prime \prime }+\left (2-2 x \right ) y^{\prime }+\left (x -2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.353 |
|
|
\[ {}x^{2} y^{\prime \prime }+6 x y^{\prime }+\left (4 x^{2}+6\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.332 |
|
|
\[ {}x y^{\prime \prime }+\left (-2 x +1\right ) y^{\prime }+\left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.339 |
|
|
\[ {}x \left (1-x \right ) y^{\prime \prime }+\left (\frac {1}{2}+2 x \right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Jacobi] |
✓ |
✓ |
0.541 |
|
|
\[ {}4 \left (t^{2}-3 t +2\right ) y^{\prime \prime }-2 y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.692 |
|
|
\[ {}2 \left (t^{2}-5 t +6\right ) y^{\prime \prime }+\left (2 t -3\right ) y^{\prime }-8 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.678 |
|
|
\[ {}3 t \left (1+t \right ) y^{\prime \prime }+t y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.721 |
|
|
\[ {}x^{2} y^{\prime \prime }+\frac {\left (x +\frac {3}{4}\right ) y}{4} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.671 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\frac {\left (x^{2}-1\right ) y}{4} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.432 |
|
|
\[ {}x y^{\prime \prime }+\left (-2 x +1\right ) y^{\prime }+\left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.383 |
|
|
\[ {}x y^{\prime \prime }-\left (1+x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Laguerre] |
✓ |
✓ |
0.39 |
|
|
\[ {}x y^{\prime \prime }+3 y^{\prime }+4 x^{3} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.491 |
|
|
\[ {}x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+2 x \left (-x^{2}+1\right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.342 |
|
|
\[ {}2 x y^{\prime \prime }+\left (x -2\right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.408 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+y x = 0 \] |
1 |
1 |
1 |
kovacic |
[_Lienard] |
✓ |
✓ |
0.313 |
|
|
\[ {}y^{\prime \prime }+2 x^{2} y^{\prime }+\left (x^{4}+2 x -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.364 |
|
|
\[ {}u^{\prime \prime }-\left (1+2 x \right ) u^{\prime }+\left (x^{2}+x -1\right ) u = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.335 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+\left (1+\frac {2}{\left (1+3 x \right )^{2}}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.7 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.358 |
|
|
\[ {}y^{\prime \prime }+\frac {2 y^{\prime }}{x}-\frac {2 y}{\left (1+x \right )^{2}} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.344 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.381 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.251 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.254 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.24 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.253 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.24 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.237 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.241 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.592 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.256 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.258 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+y x = 0 \] |
1 |
1 |
1 |
kovacic |
[_Lienard] |
✓ |
✓ |
0.353 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+3 x y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.461 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (3 x^{2}+2 x \right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.415 |
|
|
\[ {}2 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+x \left (11 x^{2}+11 x +9\right ) y^{\prime }+\left (7 x^{2}+10 x +6\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.703 |
|
|
\[ {}x y^{\prime \prime }+\left (1+x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.479 |
|
|
\[ {}x^{2} \left (x^{2}-2 x +1\right ) y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+\left (x +4\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.804 |
|
|
\[ {}2 x^{2} \left (2+x \right ) y^{\prime \prime }+5 x^{2} y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.789 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.362 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.591 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }-\left (x^{2}+\frac {5}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.494 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.266 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+4 x^{4} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.527 |
|
|
\[ {}y^{\prime \prime } = \left (x^{2}+3\right ) y \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.303 |
|
|
\[ {}y^{\prime \prime }+2 x y^{\prime }+\left (x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.232 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.276 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+\left (-8 x^{2}+4 x \right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.355 |
|
|
\[ {}y^{\prime \prime } = \left (-\frac {3}{16 x^{2}}-\frac {2}{9 \left (-1+x \right )^{2}}+\frac {3}{16 x \left (-1+x \right )}\right ) y \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.941 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +2\right ) y^{\prime }+\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.815 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }+y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
50.874 |
|
|
\[ {}\left (x^{2}-x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.705 |
|
|
\[ {}x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }-x \left (4 x^{2}+3\right ) y^{\prime }+\left (-2 x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.407 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (4 x^{6}-8 x^{5}+12 x^{4}+4 x^{3}+7 x^{2}-20 x +4\right ) y}{4 x^{4}} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.786 |
|
|
\[ {}y^{\prime \prime } = \left (\frac {6}{x^{2}}-1\right ) y \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.548 |
|
|
\[ {}y^{\prime \prime } = \left (\frac {x^{2}}{4}-\frac {11}{2}\right ) y \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.047 |
|
|
\[ {}y^{\prime \prime } = \left (\frac {1}{x}-\frac {3}{16 x^{2}}\right ) y \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.404 |
|
|
\[ {}y^{\prime \prime } = \left (-\frac {3}{16 x^{2}}-\frac {2}{9 \left (-1+x \right )^{2}}+\frac {3}{16 x \left (-1+x \right )}\right ) y \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.569 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (5 x^{2}+27\right ) y}{36 \left (x^{2}-1\right )^{2}} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
87.23 |
|
|
\[ {}y^{\prime \prime } = \left (x^{2}+3\right ) y \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.335 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.283 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.4 |
|
|
\[ {}y^{\prime }-y^{2}-y \sin \left (2 x \right )-\cos \left (2 x \right ) = 0 \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.492 |
|
|
\[ {}y^{\prime }+x y^{2}-x^{3} y-2 x = 0 \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.019 |
|
|
\[ {}x y^{\prime }-x \sqrt {x^{2}+y^{2}}-y = 0 \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
0.963 |
|
|
\[ {}\left (x -a \right ) \left (-b +x \right ) y^{\prime }+y^{2}+k \left (y+x -a \right ) \left (y+x -b \right ) = 0 \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
2.263 |
|
|
\[ {}2 x^{2} y^{\prime }-2 y^{2}-3 y x +2 x \,a^{2} = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.428 |
|
|
\[ {}3 x \left (x^{2}-1\right ) y^{\prime }+x y^{2}-\left (x^{2}+1\right ) y-3 x = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
2.325 |
|
|
\[ {}x \left (x^{3}-1\right ) y^{\prime }-2 x y^{2}+y+x^{2} = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.164 |
|
|
\[ {}\left (a \,x^{2}+b x +c \right )^{2} \left (y^{2}+y^{\prime }\right )+A = 0 \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_rational, _Riccati] |
✓ |
✓ |
3.141 |
|
|
\[ {}\left (y x +a \right ) y^{\prime }+b y = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[[_1st_order, _with_exponential_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.817 |
|
|
\[ {}y^{\prime } = \frac {\left (-1+y \ln \left (x \right )\right )^{2}}{x} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_Riccati] |
✓ |
✓ |
1.944 |
|
|
\[ {}y^{\prime } = \frac {\left (2 y \ln \left (x \right )-1\right )^{2}}{x} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_Riccati] |
✓ |
✓ |
1.98 |
|
|
\[ {}y^{\prime } = \frac {\left (4 x a -y^{2}\right )^{2}}{y} \] |
1 |
0 |
2 |
unknown |
[_rational] |
✗ |
N/A |
1.125 |
|
|
\[ {}y^{\prime } = \frac {\left (a y^{2}+b \,x^{2}\right )^{2} x}{a^{\frac {5}{2}} y} \] |
1 |
0 |
2 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.154 |
|
|
\[ {}y^{\prime } = \frac {\left (-2 y^{\frac {3}{2}}+3 \,{\mathrm e}^{x}\right )^{2} {\mathrm e}^{x}}{4 \sqrt {y}} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.749 |
|
|
\[ {}y^{\prime } = \frac {\left (1+x y^{2}\right )^{2}}{y x^{4}} \] |
1 |
0 |
2 |
unknown |
[_rational] |
✗ |
N/A |
1.073 |
|
|
\[ {}y^{\prime } = \frac {y+\sqrt {x^{2}+y^{2}}\, x^{2}}{x} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.338 |
|
|
\[ {}y^{\prime } = \frac {y+x^{3} \sqrt {x^{2}+y^{2}}}{x} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.322 |
|
|
\[ {}y^{\prime } = \frac {2 x^{2}+2 x +x^{4}-2 x^{2} y-1+y^{2}}{1+x} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
4.867 |
|
|
\[ {}y^{\prime } = \frac {2 a}{-x^{2} y+2 a y^{4} x^{2}-16 x y^{2} a^{2}+32 a^{3}} \] |
1 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
2.833 |
|
|
\[ {}y^{\prime } = \frac {x +y^{4}-2 y^{2} x^{2}+x^{4}}{y} \] |
1 |
0 |
2 |
unknown |
[_rational] |
✗ |
N/A |
1.435 |
|
|
\[ {}y^{\prime } = \frac {x}{-y+x^{4}+2 y^{2} x^{2}+y^{4}} \] |
1 |
0 |
3 |
unknown |
[_rational] |
✗ |
N/A |
1.141 |
|
|
\[ {}y^{\prime } = \frac {\left (x -y\right )^{2} \left (x +y\right )^{2} x}{y} \] |
1 |
0 |
2 |
unknown |
[_rational] |
✗ |
N/A |
1.15 |
|
|
\[ {}y^{\prime } = \frac {y \left (y+1\right )}{x \left (-y-1+y x \right )} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.508 |
|
|
\[ {}y^{\prime } = \frac {y x +y+x \sqrt {x^{2}+y^{2}}}{x \left (1+x \right )} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.56 |
|
|
\[ {}y^{\prime } = \frac {y x +y+x^{4} \sqrt {x^{2}+y^{2}}}{x \left (1+x \right )} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.672 |
|
|
\[ {}y^{\prime } = \frac {\left (2 y+1\right ) \left (y+1\right )}{x \left (-2 y-2+x +2 y x \right )} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.092 |
|
|
\[ {}y^{\prime } = \frac {b \,x^{3}+c^{2} \sqrt {a}-2 c b \,x^{2} \sqrt {a}+2 c y^{2} a^{\frac {3}{2}}+b^{2} x^{4} \sqrt {a}-2 y^{2} a^{\frac {3}{2}} b \,x^{2}+a^{\frac {5}{2}} y^{4}}{a \,x^{2} y} \] |
1 |
0 |
2 |
unknown |
[_rational] |
✗ |
N/A |
1.797 |
|
|
\[ {}y^{\prime } = \frac {y+x \sqrt {x^{2}+y^{2}}+x^{3} \sqrt {x^{2}+y^{2}}+x^{4} \sqrt {x^{2}+y^{2}}}{x} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.468 |
|
|
\[ {}y^{\prime } = \frac {4 x \left (a -1\right ) \left (a +1\right )}{4 y+y^{4} a^{2}-2 a^{4} y^{2} x^{2}+4 y^{2} a^{2} x^{2}+a^{6} x^{4}-3 a^{4} x^{4}+3 a^{2} x^{4}-y^{4}-2 y^{2} x^{2}-x^{4}} \] |
1 |
0 |
3 |
unknown |
[_rational] |
✗ |
N/A |
1.921 |
|
|
\[ {}y^{\prime } = \frac {\left (y-\ln \left (x \right )-\operatorname {Ci}\left (x \right )\right )^{2}+\cos \left (x \right )}{x} \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
2.694 |
|
|
\[ {}y^{\prime } = \frac {2 x^{2} y+x^{3}+y \ln \left (x \right ) x -y^{2}-y x}{x^{2} \left (x +\ln \left (x \right )\right )} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.816 |
|
|
\[ {}y^{\prime \prime }-\left (x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.473 |
|
|
\[ {}y^{\prime \prime }-\left (x^{2} a^{2}+a \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.509 |
|
|
\[ {}y^{\prime \prime }-\left (1+2 \tan \left (x \right )^{2}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.276 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.776 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Hermite] |
✓ |
✓ |
0.974 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.4 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-1\right ) y-{\mathrm e}^{x} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.644 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.356 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-3\right ) y-{\mathrm e}^{x^{2}} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.614 |
|
|
\[ {}y^{\prime \prime }+2 a x y^{\prime }+a^{2} y x^{2} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.254 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }+y x = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.872 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }-\left (1+x \right )^{2} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.551 |
|
|
\[ {}y^{\prime \prime }-x^{2} \left (1+x \right ) y^{\prime }+x \left (x^{4}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.616 |
|
|
\[ {}y^{\prime \prime }+x^{4} y^{\prime }-x^{3} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.988 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } \sqrt {x}+\left (\frac {1}{4 \sqrt {x}}+\frac {x}{4}-9\right ) y-x \,{\mathrm e}^{-\frac {x^{\frac {3}{2}}}{3}} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.865 |
|
|
\[ {}y^{\prime \prime }-\frac {y^{\prime }}{\sqrt {x}}+\frac {\left (x +\sqrt {x}-8\right ) y}{4 x^{2}} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.379 |
|
|
\[ {}y^{\prime \prime }-\left (2 \,{\mathrm e}^{x}+1\right ) y^{\prime }+{\mathrm e}^{2 x} y-{\mathrm e}^{3 x} = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.658 |
|
|
\[ {}y^{\prime \prime }+2 a y^{\prime } \cot \left (x a \right )+\left (-a^{2}+b^{2}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.541 |
|
|
\[ {}y^{\prime \prime }+f \left (x \right ) y^{\prime }+\left (\frac {f \left (x \right )^{2}}{4}+\frac {f^{\prime }\left (x \right )}{2}+a \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.668 |
|
|
\[ {}4 y^{\prime \prime }+4 \tan \left (x \right ) y^{\prime }-\left (5 \tan \left (x \right )^{2}+2\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.868 |
|
|
\[ {}a^{2} y^{\prime \prime }+a \left (a^{2}-2 b \,{\mathrm e}^{-x a}\right ) y^{\prime }+b^{2} {\mathrm e}^{-2 x a} y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.948 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }-y x -{\mathrm e}^{x} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.03 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+a x y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.488 |
|
|
\[ {}x y^{\prime \prime }-\left (1+x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_non_constant_coeff_transformation_on_B |
[_Laguerre] |
✓ |
✓ |
0.637 |
|
|
\[ {}x y^{\prime \prime }-\left (1+x \right ) y^{\prime }-2 \left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.467 |
|
|
\[ {}x y^{\prime \prime }-2 \left (x a +b \right ) y^{\prime }+\left (x \,a^{2}+2 a b \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.749 |
|
|
\[ {}x y^{\prime \prime }-\left (x^{2}-x -2\right ) y^{\prime }-x \left (x +3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.497 |
|
|
\[ {}x y^{\prime \prime }-\left (2 a \,x^{2}+1\right ) y^{\prime }+b \,x^{3} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.42 |
|
|
\[ {}x y^{\prime \prime }+\left (4 x^{2}-1\right ) y^{\prime }-4 x^{3} y-4 x^{5} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.033 |
|
|
\[ {}x y^{\prime \prime }+\left (2 a \,x^{3}-1\right ) y^{\prime }+\left (a^{2} x^{3}+a \right ) x^{2} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.844 |
|
|
\[ {}x y^{\prime \prime }+\left (2 a x \ln \left (x \right )+1\right ) y^{\prime }+\left (a^{2} x \ln \left (x \right )^{2}+a \ln \left (x \right )+a \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.567 |
|
|
\[ {}\left (x -3\right ) y^{\prime \prime }-\left (4 x -9\right ) y^{\prime }+\left (3 x -6\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.846 |
|
|
\[ {}\left (2 x -1\right ) y^{\prime \prime }-\left (3 x -4\right ) y^{\prime }+\left (x -3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.293 |
|
|
\[ {}4 x y^{\prime \prime }+4 y^{\prime }-\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.562 |
|
|
\[ {}a x y^{\prime \prime }+\left (b x +3 a \right ) y^{\prime }+3 b y = 0 \] |
1 |
1 |
1 |
second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.847 |
|
|
\[ {}5 \left (x a +b \right ) y^{\prime \prime }+8 a y^{\prime }+c \left (x a +b \right )^{\frac {1}{5}} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.414 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.488 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (a \,x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.61 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2} a^{2}-6\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.715 |
|
|
\[ {}x^{2} y^{\prime \prime }+a y^{\prime }-\left (b^{2} x^{2}+a b \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.084 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.132 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y-\frac {x^{2}}{\cos \left (x \right )} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.313 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y-\frac {x^{3}}{\cos \left (x \right )} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.803 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2} a^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.679 |
|
|
\[ {}x^{2} y^{\prime \prime }+x^{2} y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.475 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2}-1\right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.492 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }+\left (x -9\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.562 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }+\left (3 x -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.529 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (-1+x \right ) y^{\prime }+\left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.754 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (x^{2}-2 x \right ) y^{\prime }-\left (2+3 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.042 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (x +4\right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.293 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1+2 x \right ) y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.546 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x \left (1+x \right ) y^{\prime }+2 \left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.708 |
|
|
\[ {}x^{2} y^{\prime \prime }+a \,x^{2} y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.532 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (a +2 b \right ) x^{2} y^{\prime }+\left (\left (a +b \right ) b \,x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.746 |
|
|
\[ {}x^{2} y^{\prime \prime }+x^{3} y^{\prime }+\left (x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.826 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x^{2}+2\right ) y^{\prime }+\left (x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.172 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x^{3} y^{\prime }+\left (4 x^{4}+2 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.844 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.876 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.73 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+3 x y^{\prime }+a y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.543 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-\left (1+3 x \right ) y^{\prime }-\left (x^{2}-x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.602 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+4 x y^{\prime }+\left (x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.358 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+2 a x y^{\prime }+a \left (a -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
1.227 |
|
|
\[ {}\left (-a^{2}+x^{2}\right ) y^{\prime \prime }+8 x y^{\prime }+12 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.842 |
|
|
\[ {}x \left (1+x \right ) y^{\prime \prime }-\left (-1+x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.886 |
|
|
\[ {}\left (x^{2}+x -2\right ) y^{\prime \prime }+\left (x^{2}-x \right ) y^{\prime }-\left (6 x^{2}+7 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.085 |
|
|
\[ {}\left (1+x \right )^{2} y^{\prime \prime }+\left (x^{2}+x -1\right ) y^{\prime }-\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.238 |
|
|
\[ {}\left (x^{2}+3 x +4\right ) y^{\prime \prime }+\left (x^{2}+x +1\right ) y^{\prime }-\left (2 x +3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.384 |
|
|
\[ {}\left (2 x^{2}+6 x +4\right ) y^{\prime \prime }+\left (10 x^{2}+21 x +8\right ) y^{\prime }+\left (12 x^{2}+17 x +8\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.493 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }-\left (4 x^{2}+1\right ) y-4 \sqrt {x^{3}}\, {\mathrm e}^{x} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.674 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }-\left (a \,x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.567 |
|
|
|
|||||||||
|
\[ {}4 x^{2} y^{\prime \prime }+8 x y^{\prime }-\left (4 x^{2}+12 x +3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.023 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x \left (2 x -1\right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.555 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x^{3} y^{\prime }+\left (x^{2}+6\right ) \left (x^{2}-4\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.605 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x^{2} \ln \left (x \right ) y^{\prime }+\left (x^{2} \ln \left (x \right )^{2}+2 x -8\right ) y-4 x^{2} \sqrt {{\mathrm e}^{x} x^{-x}} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.78 |
|
|
\[ {}x \left (4 x -1\right ) y^{\prime \prime }+\left (\left (4 a +2\right ) x -a \right ) y^{\prime }+a \left (a -1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.119 |
|
|
\[ {}9 x \left (-1+x \right ) y^{\prime \prime }+3 \left (2 x -1\right ) y^{\prime }-20 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Jacobi] |
✓ |
✓ |
0.947 |
|
|
\[ {}16 x^{2} y^{\prime \prime }+\left (3+4 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.462 |
|
|
\[ {}16 x^{2} y^{\prime \prime }+32 x y^{\prime }-\left (5+4 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.582 |
|
|
\[ {}50 x \left (-1+x \right ) y^{\prime \prime }+25 \left (2 x -1\right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[_Jacobi, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.766 |
|
|
\[ {}\left (x^{2} a^{2}-1\right ) y^{\prime \prime }+2 a^{2} x y^{\prime }-2 a^{2} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[_Gegenbauer] |
✓ |
✓ |
0.905 |
|
|
\[ {}x^{3} y^{\prime \prime }+x y^{\prime }-\left (2 x +3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.53 |
|
|
\[ {}x^{3} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.324 |
|
|
\[ {}x \left (x^{2}+1\right ) y^{\prime \prime }+2 \left (x^{2}-1\right ) y^{\prime }-2 y x = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.878 |
|
|
\[ {}x \left (x^{2}+2\right ) y^{\prime \prime }-y^{\prime }-6 y x = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.199 |
|
|
\[ {}x \left (x^{2}-2\right ) y^{\prime \prime }-\left (x^{3}+3 x^{2}-2 x -2\right ) y^{\prime }+\left (x^{2}+4 x +2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.996 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (1+2 x \right ) y^{\prime }+\left (1+2 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.858 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (5 x -4\right ) y^{\prime }}{x \left (-1+x \right )}-\frac {\left (9 x -6\right ) y}{x^{2} \left (-1+x \right )} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.414 |
|
|
\[ {}y^{\prime \prime } = -\frac {y^{\prime }}{1+x}-\frac {y}{x \left (1+x \right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.534 |
|
|
\[ {}y^{\prime \prime } = \frac {2 y}{x \left (-1+x \right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.842 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (x -4\right ) y^{\prime }}{2 x \left (x -2\right )}-\frac {\left (x -3\right ) y}{2 x^{2} \left (x -2\right )} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.77 |
|
|
\[ {}y^{\prime \prime } = \frac {y^{\prime }}{1+x}-\frac {\left (1+3 x \right ) y}{4 x^{2} \left (1+x \right )} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.175 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (1-3 x \right ) y}{\left (-1+x \right ) \left (2 x -1\right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.968 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (6 x -1\right ) y^{\prime }}{3 x \left (x -2\right )}+\frac {y}{3 x^{2} \left (x -2\right )} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.823 |
|
|
\[ {}y^{\prime \prime } = \frac {2 \left (x a +2 b \right ) y^{\prime }}{x \left (x a +b \right )}-\frac {\left (2 x a +6 b \right ) y}{\left (x a +b \right ) x^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.826 |
|
|
\[ {}y^{\prime \prime } = -\frac {a y}{x^{4}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.362 |
|
|
\[ {}y^{\prime \prime } = -\frac {y^{\prime }}{x^{3}}+\frac {2 y}{x^{4}} \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.706 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (a +b \right ) y^{\prime }}{x^{2}}-\frac {\left (x \left (a +b \right )+a b \right ) y}{x^{4}} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.586 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x^{3}}+\frac {y}{x^{4}} \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.421 |
|
|
\[ {}y^{\prime \prime } = -\frac {2 \left (x +a \right ) y^{\prime }}{x^{2}}-\frac {b y}{x^{4}} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.636 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (2 x^{2}-1\right ) y^{\prime }}{x^{3}}-\frac {y}{x^{4}} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.224 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (2 x^{2}-1\right ) y^{\prime }}{x^{3}}-\frac {2 y}{x^{4}} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.299 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (x^{3}-1\right ) y^{\prime }}{x \left (x^{3}+1\right )}+\frac {x y}{x^{3}+1} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.591 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (x^{2}-2\right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {\left (x^{2}-2\right ) y}{x^{2} \left (x^{2}-1\right )} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.028 |
|
|
\[ {}y^{\prime \prime } = \frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (a \left (a +1\right )-a \,x^{2} \left (a +3\right )\right ) y}{x^{2} \left (x^{2}-1\right )} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
51.416 |
|
|
\[ {}y^{\prime \prime } = -\frac {a y}{\left (x^{2}+1\right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[_Halm] |
✓ |
✓ |
0.98 |
|
|
\[ {}y^{\prime \prime } = -\frac {a y}{\left (x^{2}-1\right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.086 |
|
|
\[ {}y^{\prime \prime } = -\frac {b^{2} y}{\left (a^{2}+x^{2}\right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.382 |
|
|
\[ {}y^{\prime \prime } = -\frac {2 \left (x^{2}-1\right ) y^{\prime }}{x \left (-1+x \right )^{2}}-\frac {\left (-2 x^{2}+2 x +2\right ) y}{x^{2} \left (-1+x \right )^{2}} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.443 |
|
|
\[ {}y^{\prime \prime } = \frac {12 y}{\left (1+x \right )^{2} \left (x^{2}+2 x +3\right )} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.45 |
|
|
\[ {}y^{\prime \prime } = -\frac {b y}{x^{2} \left (x -a \right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.204 |
|
|
\[ {}y^{\prime \prime } = -\frac {b y}{x^{2} \left (x -a \right )^{2}}+c \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
47.664 |
|
|
\[ {}y^{\prime \prime } = \frac {c y}{\left (x -a \right )^{2} \left (-b +x \right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.251 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (\left (\alpha +\beta +1\right ) \left (x -a \right )^{2} \left (-b +x \right )+\left (1-\alpha -\beta \right ) \left (-b +x \right )^{2} \left (x -a \right )\right ) y^{\prime }}{\left (x -a \right )^{2} \left (-b +x \right )^{2}}-\frac {\alpha \beta \left (-b +a \right )^{2} y}{\left (x -a \right )^{2} \left (-b +x \right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.905 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (a \,x^{2}+a -3\right ) y}{4 \left (x^{2}+1\right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[_Halm] |
✓ |
✓ |
4.079 |
|
|
\[ {}y^{\prime \prime } = \frac {18 y}{\left (1+2 x \right )^{2} \left (x^{2}+x +1\right )} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.473 |
|
|
\[ {}y^{\prime \prime } = \frac {3 y}{4 \left (x^{2}+x +1\right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.487 |
|
|
\[ {}y^{\prime \prime } = -\frac {3 y}{16 x^{2} \left (-1+x \right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.759 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (7 a \,x^{2}+5\right ) y^{\prime }}{x \left (a \,x^{2}+1\right )}-\frac {\left (15 a \,x^{2}+5\right ) y}{x^{2} \left (a \,x^{2}+1\right )} \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.726 |
|
|
\[ {}y^{\prime \prime } = -\frac {2 y^{\prime }}{x}-\frac {c y}{x^{2} \left (x a +b \right )^{2}} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.232 |
|
|
\[ {}y^{\prime \prime } = -\frac {y}{\left (x a +b \right )^{4}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.736 |
|
|
\[ {}y^{\prime \prime } = -\frac {A y}{\left (a \,x^{2}+b x +c \right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
3.127 |
|
|
\[ {}y^{\prime \prime } = -\frac {y^{\prime }}{x^{4}}+\frac {y}{x^{5}} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.875 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (1+3 x \right ) y^{\prime }}{\left (-1+x \right ) \left (1+x \right )}-\frac {36 \left (1+x \right )^{2} y}{\left (-1+x \right )^{2} \left (5+3 x \right )^{2}} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.056 |
|
|
\[ {}y^{\prime \prime } = \frac {y^{\prime }}{x}-\frac {a y}{x^{6}} \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.235 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (3 x^{2}+a \right ) y^{\prime }}{x^{3}}-\frac {b y}{x^{6}} \] |
1 |
1 |
1 |
second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.732 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {\left (-2 x^{2}+1\right ) y}{4 x^{6}} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.326 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (2 x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {\left (a \,x^{4}+10 x^{2}+1\right ) y}{4 x^{6}} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.447 |
|
|
\[ {}y^{\prime \prime } = -\frac {27 x y}{16 \left (x^{3}-1\right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
77.625 |
|
|
\[ {}y^{\prime \prime } = \frac {y^{\prime }}{x \left (-1+\ln \left (x \right )\right )}-\frac {y}{x^{2} \left (-1+\ln \left (x \right )\right )} \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.776 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (\sin \left (x \right ) x^{2}-2 x \cos \left (x \right )\right ) y^{\prime }}{x^{2} \cos \left (x \right )}-\frac {\left (2 \cos \left (x \right )-x \sin \left (x \right )\right ) y}{x^{2} \cos \left (x \right )} \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.674 |
|
|
\[ {}y^{\prime \prime } = -\frac {a \left (n -1\right ) \sin \left (2 x a \right ) y^{\prime }}{\cos \left (x a \right )^{2}}-\frac {n \,a^{2} \left (\left (n -1\right ) \sin \left (x a \right )^{2}+\cos \left (x a \right )^{2}\right ) y}{\cos \left (x a \right )^{2}} \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.033 |
|
|
\[ {}y^{\prime \prime } = \frac {2 y}{\sin \left (x \right )^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.761 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (-a^{2} \cos \left (x \right )^{2}-\left (3-2 a \right ) \cos \left (x \right )-3+3 a \right ) y}{\sin \left (x \right )^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.692 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (-\left (a^{2} b^{2}-\left (a +1\right )^{2}\right ) \sin \left (x \right )^{2}-a \left (a +1\right ) b \sin \left (2 x \right )-a \left (a -1\right )\right ) y}{\sin \left (x \right )^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.293 |
|
|
\[ {}y^{\prime \prime } = \frac {\cos \left (2 x \right ) y^{\prime }}{\sin \left (2 x \right )}-2 y \] |
1 |
1 |
1 |
second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
8.474 |
|
|
\[ {}y^{\prime \prime } = -\frac {\cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}-\frac {\left (-17 \sin \left (x \right )^{2}-1\right ) y}{4 \sin \left (x \right )^{2}} \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.046 |
|
|
\[ {}y^{\prime \prime } = -\frac {\sin \left (x \right ) y^{\prime }}{\cos \left (x \right )}-\frac {\left (2 x^{2}+x^{2} \sin \left (x \right )^{2}-24 \cos \left (x \right )^{2}\right ) y}{4 x^{2} \cos \left (x \right )^{2}}+\sqrt {\cos \left (x \right )} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
5.937 |
|
|
\[ {}y^{\prime \prime } = -\frac {y^{\prime }}{x}-\frac {\left (-1+x \right ) y}{x^{4}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.414 |
|
|
\[ {}y^{\prime \prime } = -\frac {y^{\prime }}{x}-\frac {\left (-x -1\right ) y}{x^{4}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.34 |
|
|
\[ {}y^{\prime \prime } = -\frac {b^{2} y}{\left (-a^{2}+x^{2}\right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.409 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }-4 x^{2} y^{\prime \prime }+\left (x^{2}+8\right ) x y^{\prime }-2 \left (x^{2}+4\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.264 |
|
|
\[ {}y^{\prime \prime \prime \prime } \sin \left (x \right )^{6}+4 y^{\prime \prime \prime } \sin \left (x \right )^{5} \cos \left (x \right )-6 y^{\prime \prime } \sin \left (x \right )^{6}-4 y^{\prime } \sin \left (x \right )^{5} \cos \left (x \right )+y \sin \left (x \right )^{6}-f = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
0.101 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (2 a +b -1\right ) x y^{\prime }+\left (c^{2} b^{2} x^{2 b}+a \left (a +b \right )\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.25 |
|
|
\[ {}8 \left (-x^{3}+1\right ) y y^{\prime \prime }-4 \left (-x^{3}+1\right ) {y^{\prime }}^{2}-12 x^{2} y y^{\prime }+3 x y^{2} = 0 \] |
1 |
0 |
2 |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.034 |
|
|
\[ {}y^{\prime } = y^{2}-x^{2} a^{2}+3 a \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.263 |
|
|
\[ {}y^{\prime } = y^{2}+a n \,x^{n -1}-a^{2} x^{2 n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
82.267 |
|
|
\[ {}y^{\prime } = a \,x^{n} y^{2}+b m \,x^{m -1}-a \,b^{2} x^{n +2 m} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
84.497 |
|
|
\[ {}x^{2} y^{\prime } = y^{2} x^{2}-a^{2} x^{4}+a \left (1-2 b \right ) x^{2}-b \left (b +1\right ) \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.595 |
|
|
\[ {}\left (a \,x^{2}+b x +c \right )^{2} \left (y^{2}+y^{\prime }\right )+A = 0 \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_rational, _Riccati] |
✓ |
✓ |
3.171 |
|
|
\[ {}\left (a \,x^{n}+b \right ) y^{\prime } = b y^{2}+a \,x^{n -2} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
2.296 |
|
|
\[ {}2 x^{2} y^{\prime } = 2 y^{2}+3 y x -2 x \,a^{2} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.289 |
|
|
\[ {}\left (a \,x^{2}+b x +c \right ) y^{\prime } = y^{2}+\left (2 \lambda x +b \right ) y+\lambda \left (\lambda -a \right ) x^{2}+\mu \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
3.106 |
|
|
\[ {}\left (a \,x^{2}+b x +c \right ) y^{\prime } = y^{2}+\left (x a +\mu \right ) y-\lambda ^{2} x^{2}+\lambda \left (b -\mu \right ) x +c \lambda \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✗ |
102.457 |
|
|
\[ {}\left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime } = y^{2}+\left (a_{1} x +b_{1} \right ) y-\lambda \left (\lambda +a_{1} -a_{2} \right ) x^{2}+\lambda \left (b_{2} -b_{1} \right ) x +\lambda c_{2} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✗ |
96.188 |
|
|
\[ {}\left (x -a \right ) \left (-b +x \right ) y^{\prime }+y^{2}+k \left (y+x -a \right ) \left (y+x -b \right ) = 0 \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
2.294 |
|
|
\[ {}y^{\prime } = y^{2}+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} {\mathrm e}^{2 \lambda x} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.615 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{\lambda x} y^{2}+a \,{\mathrm e}^{x \mu } y+a \lambda \,{\mathrm e}^{\left (\mu -\lambda \right ) x} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.493 |
|
|
\[ {}y^{\prime } = a \,{\mathrm e}^{x \mu } y^{2}+a b \,{\mathrm e}^{x \left (\lambda +\mu \right )} y-b \lambda \,{\mathrm e}^{\lambda x} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
87.883 |
|
|
\[ {}y^{\prime } = y^{2}-a^{2}+a \lambda \sinh \left (\lambda x \right )-a^{2} \sinh \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
9.12 |
|
|
\[ {}y^{\prime } = \lambda \sinh \left (\lambda x \right ) y^{2}-\lambda \sinh \left (\lambda x \right )^{3} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
3.727 |
|
|
\[ {}y^{\prime } = \left (a \sinh \left (\lambda x \right )^{2}-\lambda \right ) y^{2}-a \sinh \left (\lambda x \right )^{2}+\lambda -a \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
6.715 |
|
|
\[ {}y^{\prime } = \left (a \cosh \left (\lambda x \right )^{2}-\lambda \right ) y^{2}+a +\lambda -a \cosh \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
6.763 |
|
|
\[ {}2 y^{\prime } = \left (a -\lambda +a \cosh \left (\lambda x \right )\right ) y^{2}+a +\lambda -a \cosh \left (\lambda x \right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
6.428 |
|
|
\[ {}y^{\prime } = y^{2}+a \lambda -a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.293 |
|
|
\[ {}y^{\prime } = y^{2}+3 a \lambda -\lambda ^{2}-a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.572 |
|
|
\[ {}y^{\prime } = y^{2}+a \lambda -a \left (a +\lambda \right ) \coth \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
6.882 |
|
|
\[ {}y^{\prime } = y^{2}-\lambda ^{2}+3 a \lambda -a \left (a +\lambda \right ) \coth \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
5.214 |
|
|
\[ {}y^{\prime } = y^{2}-2 \lambda ^{2} \tanh \left (\lambda x \right )^{2}-2 \lambda ^{2} \coth \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
17.702 |
|
|
\[ {}y^{\prime } = y^{2}+a \lambda +b \lambda -2 a b -a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}-b \left (b +\lambda \right ) \coth \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
17.091 |
|
|
\[ {}y^{\prime } = y^{2}-a^{2}+a \lambda \sin \left (\lambda x \right )+a^{2} \sin \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.316 |
|
|
\[ {}y^{\prime } = \lambda \sin \left (\lambda x \right ) y^{2}+\lambda \sin \left (\lambda x \right )^{3} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.904 |
|
|
\[ {}2 y^{\prime } = \left (\lambda +a -a \sin \left (\lambda x \right )\right ) y^{2}+\lambda -a -a \sin \left (\lambda x \right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✗ |
166.727 |
|
|
\[ {}y^{\prime } = \left (\lambda +a \sin \left (\lambda x \right )^{2}\right ) y^{2}+\lambda -a +a \sin \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
7.664 |
|
|
\[ {}y^{\prime } = y^{2}-a^{2}+a \lambda \cos \left (\lambda x \right )+a^{2} \cos \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.056 |
|
|
\[ {}y^{\prime } = \lambda \cos \left (\lambda x \right ) y^{2}+\lambda \cos \left (\lambda x \right )^{3} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
14.6 |
|
|
\[ {}2 y^{\prime } = \left (\lambda +a -\cos \left (\lambda x \right ) a \right ) y^{2}+\lambda -a -\cos \left (\lambda x \right ) a \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
19.349 |
|
|
\[ {}y^{\prime } = \left (\lambda +a \cos \left (\lambda x \right )^{2}\right ) y^{2}+\lambda -a +a \cos \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
8.007 |
|
|
\[ {}y^{\prime } = y^{2}+a \lambda +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.568 |
|
|
\[ {}y^{\prime } = y^{2}+\lambda ^{2}+3 a \lambda +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
5.586 |
|
|
\[ {}y^{\prime } = y^{2}+a \lambda +a \left (\lambda -a \right ) \cot \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
6.339 |
|
|
\[ {}y^{\prime } = y^{2}+\lambda ^{2}+3 a \lambda +a \left (\lambda -a \right ) \cot \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
6.469 |
|
|
\[ {}\sin \left (2 x \right )^{n +1} y^{\prime } = a y^{2} \sin \left (x \right )^{2 n}+b \cos \left (x \right )^{2 n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
17.338 |
|
|
\[ {}y^{\prime } = y^{2}+a \lambda +b \lambda +2 a b +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2}+b \left (\lambda -b \right ) \cot \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
7.128 |
|
|
\[ {}y^{\prime \prime }-\left (x^{2} a^{2}+a \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.556 |
|
|
\[ {}y^{\prime \prime }+a^{3} x \left (-x a +2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.457 |
|
|
\[ {}y^{\prime \prime }-a \left (a \,x^{2 n}+n \,x^{n -1}\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.369 |
|
|
\[ {}y^{\prime \prime }-a \,x^{n -2} \left (a \,x^{n}+n +1\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.336 |
|
|
\[ {}y^{\prime \prime }+\left (x a +b \right ) y^{\prime }-a y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.933 |
|
|
\[ {}y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+c \left (x a +b -c \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.459 |
|
|
\[ {}y^{\prime \prime }+\left (x a +2 b \right ) y^{\prime }+\left (a b x +b^{2}-a \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.453 |
|
|
\[ {}y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{2}+b x +1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.823 |
|
|
\[ {}y^{\prime \prime }+2 \left (x a +b \right ) y^{\prime }+\left (x^{2} a^{2}+2 a b x +c \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.857 |
|
|
\[ {}y^{\prime \prime }+a \left (-b^{2}+x^{2}\right ) y^{\prime }-a \left (x +b \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.863 |
|
|
\[ {}y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c \left (a \,x^{2}+b -c \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.796 |
|
|
\[ {}y^{\prime \prime }+\left (a \,x^{2}+2 b \right ) y^{\prime }+\left (a b \,x^{2}-x a +b^{2}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.172 |
|
|
\[ {}y^{\prime \prime }+\left (2 x^{2}+a \right ) y^{\prime }+\left (x^{4}+a \,x^{2}+b +2 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.693 |
|
|
\[ {}y^{\prime \prime }+\left (a b \,x^{2}+b x +2 a \right ) y^{\prime }+a^{2} \left (b \,x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.973 |
|
|
\[ {}y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y^{\prime }+x \left (a b \,x^{2}+b c +2 a \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.945 |
|
|
\[ {}y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (a b \,x^{3}+a c \,x^{2}+b \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.831 |
|
|
\[ {}y^{\prime \prime }+\left (a \,x^{3}+2 b \right ) y^{\prime }+\left (a b \,x^{3}-a \,x^{2}+b^{2}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.733 |
|
|
\[ {}y^{\prime \prime }+\left (a \,x^{3}+b x \right ) y^{\prime }+2 \left (2 a \,x^{2}+b \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.34 |
|
|
\[ {}y^{\prime \prime }+\left (a b \,x^{3}+b \,x^{2}+2 a \right ) y^{\prime }+a^{2} \left (b \,x^{3}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.941 |
|
|
\[ {}y^{\prime \prime }+2 a \,x^{n} y^{\prime }+a \left (a \,x^{2 n}+n \,x^{n -1}\right ) y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.984 |
|
|
\[ {}y^{\prime \prime }+\left (a \,x^{n}+2 b \right ) y^{\prime }+\left (a b \,x^{n}-a \,x^{n -1}+b^{2}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.644 |
|
|
\[ {}x y^{\prime \prime }+\left (1-3 n \right ) y^{\prime }-a^{2} n^{2} x^{2 n -1} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_Emden, _Fowler]] |
✗ |
N/A |
0.341 |
|
|
\[ {}x y^{\prime \prime }+a y^{\prime }+b \,x^{n} \left (-b \,x^{n +1}+a +n \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.834 |
|
|
\[ {}x y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+c \left (\left (a -c \right ) x +b \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.906 |
|
|
\[ {}x y^{\prime \prime }+\left (2 x a +b \right ) y^{\prime }+a \left (x a +b \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.698 |
|
|
\[ {}x y^{\prime \prime }-\left (x a +1\right ) y^{\prime }-b \,x^{2} \left (b x +a \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.813 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x a +1\right ) y^{\prime }+\left (b \,x^{3}+x \,a^{2}+a \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.893 |
|
|
\[ {}x y^{\prime \prime }+\left (a b \,x^{2}+b -5\right ) y^{\prime }+2 a^{2} \left (b -2\right ) x^{3} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.498 |
|
|
\[ {}x y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }-\left (a c \,x^{2}+\left (b c +c^{2}+a \right ) x +b +2 c \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.06 |
|
|
\[ {}x y^{\prime \prime }+\left (a \,x^{2}+b x +2\right ) y^{\prime }+b y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.925 |
|
|
\[ {}x y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (c -1\right ) \left (x a +b \right ) y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.041 |
|
|
\[ {}x y^{\prime \prime }+\left (a \,x^{3}+b \right ) y^{\prime }+a \left (b -1\right ) x^{2} y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.554 |
|
|
\[ {}x y^{\prime \prime }+\left (a \,x^{3}+b \,x^{2}+2\right ) y^{\prime }+b x y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.227 |
|
|
\[ {}x y^{\prime \prime }+\left (a b \,x^{3}+b \,x^{2}+x a -1\right ) y^{\prime }+a^{2} b \,x^{3} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.143 |
|
|
\[ {}x y^{\prime \prime }+\left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime }+\left (d -1\right ) \left (a \,x^{2}+b x +c \right ) y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.665 |
|
|
\[ {}x y^{\prime \prime }+\left (a \,x^{n}+2\right ) y^{\prime }+a \,x^{n -1} y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.571 |
|
|
\[ {}x y^{\prime \prime }+\left (x^{n}+1-n \right ) y^{\prime }+b \,x^{2 n -1} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.415 |
|
|
\[ {}x y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+a \left (b -1\right ) x^{n -1} y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.816 |
|
|
\[ {}x y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+a \left (b +n -1\right ) x^{n -1} y = 0 \] |
1 |
1 |
1 |
second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.223 |
|
|
\[ {}\left (x a +b \right ) y^{\prime \prime }+s \left (c x +d \right ) y^{\prime }-s^{2} \left (\left (a +c \right ) x +b +d \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.576 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (x^{2} a^{2}+2 a b x +b^{2}-b \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.785 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (a \,x^{3}+\frac {5}{16}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.441 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (a^{2} x^{4}+a \left (2 b -1\right ) x^{2}+b \left (b +1\right )\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
12.559 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (a^{2} x^{2 n}+a \left (2 b +n -1\right ) x^{n}+b \left (b -1\right )\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.655 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }-\left (x^{2} a^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.663 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 a x y^{\prime }+\left (b^{2} x^{2}+a \left (a +1\right )\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.678 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 a x y^{\prime }+\left (-b^{2} x^{2}+a \left (a +1\right )\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.648 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{2}+b \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.867 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }-b y = 0 \] |
1 |
1 |
1 |
second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.384 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (k \left (a -k \right ) x^{2}+\left (a n +b k -2 k n \right ) x +n \left (b -n -1\right )\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.269 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (a \,x^{2}+\left (a b -1\right ) x +b \right ) y^{\prime }+a^{2} b x y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.265 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (a \,x^{n}+b \right ) y^{\prime }+b \left (a \,x^{n}-1\right ) y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.917 |
|
|
\[ {}\left (-a^{2}+x^{2}\right ) y^{\prime \prime }+b y^{\prime }-6 y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.917 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-3 x y^{\prime }+n \left (n +2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.979 |
|
|
\[ {}\left (-a^{2}+x^{2}\right ) y^{\prime \prime }+2 b x y^{\prime }+b \left (b -1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
63.584 |
|
|
\[ {}\left (a^{2}+x^{2}\right ) y^{\prime \prime }+2 b x y^{\prime }+b \left (b -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.881 |
|
|
\[ {}\left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (c \,x^{2}+d \right ) y^{\prime }+\lambda \left (\left (-a \lambda +c \right ) x^{2}+d -b \lambda \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.89 |
|
|
\[ {}\left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (\lambda \left (a +c \right ) x^{2}+\left (c -a \right ) x +2 b \lambda \right ) y^{\prime }+\lambda ^{2} \left (c \,x^{2}+b \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.071 |
|
|
\[ {}\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (k x +d \right ) y^{\prime }-k y = 0 \] |
1 |
1 |
1 |
second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
68.987 |
|
|
\[ {}\left (a \,x^{2}+2 b x +c \right ) y^{\prime \prime }+3 \left (x a +b \right ) y^{\prime }+d y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.694 |
|
|
\[ {}\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }-\left (-k^{2}+x^{2}\right ) y^{\prime }+\left (x +k \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
4.032 |
|
|
\[ {}\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (k^{3}+x^{3}\right ) y^{\prime }-\left (k^{2}-k x +x^{2}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.861 |
|
|
\[ {}x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+b y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.75 |
|
|
\[ {}x^{3} y^{\prime \prime }+\left (a \,x^{3}+a b x -x^{2}+b \right ) y^{\prime }+a^{2} b x y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.777 |
|
|
\[ {}x \left (a \,x^{2}+b \right ) y^{\prime \prime }+2 \left (a \,x^{2}+b \right ) y^{\prime }-2 a x y = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.693 |
|
|
\[ {}x^{2} \left (x a +b \right ) y^{\prime \prime }+\left (c \,x^{2}+\left (a \lambda +2 b \right ) x +b \lambda \right ) y^{\prime }+\lambda \left (c -2 a \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.437 |
|
|
\[ {}x^{2} \left (x a +b \right ) y^{\prime \prime }-2 x \left (x a +2 b \right ) y^{\prime }+2 \left (x a +3 b \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.118 |
|
|
\[ {}x^{2} \left (x a +b \right ) y^{\prime \prime }+\left (a \left (2-n -m \right ) x^{2}-b \left (m +n \right ) x \right ) y^{\prime }+\left (a m \left (n -1\right ) x +b n \left (1+m \right )\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.094 |
|
|
\[ {}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +2 c \right ) y^{\prime }+\left (\beta -2 b \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
4.786 |
|
|
\[ {}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +2 c \right ) y^{\prime }-\left (\alpha x +2 b -\beta \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.704 |
|
|
\[ {}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (-2 a \,x^{2}-\left (b +1\right ) x +k \right ) y^{\prime }+2 \left (x a +1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
5.555 |
|
|
\[ {}\left (a \,x^{3}+x^{2}+b \right ) y^{\prime \prime }+a^{2} x \left (x^{2}-b \right ) y^{\prime }-a^{3} b x y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
6.948 |
|
|
\[ {}2 x \left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (a \,x^{2}-c \right ) y^{\prime }+\lambda \,x^{2} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
211.085 |
|
|
\[ {}\left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }-\left (-\lambda ^{2}+x^{2}\right ) y^{\prime }+\left (x +\lambda \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
192.23 |
|
|
\[ {}2 \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+\left (3 a \,x^{2}+2 b x +c \right ) y^{\prime }+\lambda y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✗ |
57.526 |
|
|
\[ {}2 \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+3 \left (3 a \,x^{2}+2 b x +c \right ) y^{\prime }+\left (6 x a +2 b +\lambda \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
7.718 |
|
|
\[ {}\left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+\left (\lambda ^{3}+x^{3}\right ) y^{\prime }-\left (\lambda ^{2}-\lambda x +x^{2}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
351.694 |
|
|
\[ {}x^{4} y^{\prime \prime }+a y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.546 |
|
|
\[ {}x^{4} y^{\prime \prime }-\left (a +b \right ) x^{2} y^{\prime }+\left (x \left (a +b \right )+a b \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.506 |
|
|
\[ {}x^{4} y^{\prime \prime }+2 x^{2} \left (x +a \right ) y^{\prime }+b y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.898 |
|
|
\[ {}x^{2} \left (x -a \right )^{2} y^{\prime \prime }+b y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.21 |
|
|
\[ {}x^{2} \left (x -a \right )^{2} y^{\prime \prime }+b y = c \,x^{2} \left (x -a \right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
7.906 |
|
|
\[ {}\left (x^{2}+1\right )^{2} y^{\prime \prime }+a y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[_Halm] |
✓ |
✓ |
0.826 |
|
|
\[ {}\left (x^{2}-1\right )^{2} y^{\prime \prime }+a y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.049 |
|
|
\[ {}\left (a^{2}+x^{2}\right )^{2} y^{\prime \prime }+b^{2} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.395 |
|
|
\[ {}\left (-a^{2}+x^{2}\right )^{2} y^{\prime \prime }+b^{2} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.463 |
|
|
\[ {}4 \left (x^{2}+1\right )^{2} y^{\prime \prime }+\left (a \,x^{2}+a -3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[_Halm] |
✓ |
✓ |
3.759 |
|
|
|
|||||||||
|
\[ {}\left (a \,x^{2}+b \right )^{2} y^{\prime \prime }+\left (2 x a +c \right ) \left (a \,x^{2}+b \right ) y^{\prime }+k y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.025 |
|
|
\[ {}\left (a \,x^{2}+b \right )^{2} y^{\prime \prime }+\left (a \,x^{2}+b \right ) \left (c \,x^{2}+d \right ) y^{\prime }+2 \left (-a d +b c \right ) x y = 0 \] |
1 |
1 |
1 |
second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.494 |
|
|
\[ {}\left (x -a \right )^{2} \left (-b +x \right )^{2} y^{\prime \prime }-c y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.163 |
|
|
\[ {}\left (x -a \right )^{2} \left (-b +x \right )^{2} y^{\prime \prime }+\left (x -a \right ) \left (-b +x \right ) \left (2 x +\lambda \right ) y^{\prime }+\mu y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.196 |
|
|
\[ {}\left (a \,x^{2}+b x +c \right )^{2} y^{\prime \prime }+y A = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.625 |
|
|
\[ {}\left (a \,x^{2}+b x +c \right )^{2} y^{\prime \prime }+\left (2 x a +k \right ) \left (a \,x^{2}+b x +c \right ) y^{\prime }+m y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
4.23 |
|
|
\[ {}x^{6} y^{\prime \prime }-x^{5} y^{\prime }+a y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.246 |
|
|
\[ {}x^{6} y^{\prime \prime }+\left (3 x^{2}+a \right ) x^{3} y^{\prime }+b y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.4 |
|
|
\[ {}x \left (x^{n}+1\right ) y^{\prime \prime }+\left (\left (-b +a \right ) x^{n}+a -n \right ) y^{\prime }+b \left (-a +1\right ) x^{n -1} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.422 |
|
|
\[ {}\left (x^{n}+a \right )^{2} y^{\prime \prime }-b \,x^{n -2} \left (\left (b -1\right ) x^{n}+a \left (n -1\right )\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
80.737 |
|
|
\[ {}x^{2} \left (a \,x^{n}+b \right )^{2} y^{\prime \prime }+\left (n +1\right ) x \left (a^{2} x^{2 n}-b^{2}\right ) y^{\prime }+c y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.321 |
|
|
\[ {}y^{\prime \prime }+a \left (\lambda \,{\mathrm e}^{\lambda x}-a \,{\mathrm e}^{2 \lambda x}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.134 |
|
|
\[ {}y^{\prime \prime }-\left (a^{2} {\mathrm e}^{2 x}+a \left (2 b +1\right ) {\mathrm e}^{x}+b^{2}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.197 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime }+b \,{\mathrm e}^{2 x a} y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode_form_A |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.17 |
|
|
\[ {}y^{\prime \prime }+2 a \,{\mathrm e}^{\lambda x} y^{\prime }+a \,{\mathrm e}^{\lambda x} \left ({\mathrm e}^{\lambda x} a +\lambda \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.806 |
|
|
\[ {}y^{\prime \prime }+\left (a +b \right ) {\mathrm e}^{\lambda x} y^{\prime }+a \,{\mathrm e}^{\lambda x} \left (b \,{\mathrm e}^{\lambda x}+\lambda \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.374 |
|
|
\[ {}y^{\prime \prime }-\left (a +2 b \,{\mathrm e}^{x a}\right ) y^{\prime }+b^{2} {\mathrm e}^{2 x a} y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.495 |
|
|
\[ {}y^{\prime \prime }+\left (a \,{\mathrm e}^{2 \lambda x}+\lambda \right ) y^{\prime }-a \lambda \,{\mathrm e}^{2 \lambda x} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.48 |
|
|
\[ {}y^{\prime \prime }+\left ({\mathrm e}^{\lambda x} a +b \right ) y^{\prime }+c \left ({\mathrm e}^{\lambda x} a +b -c \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.41 |
|
|
\[ {}y^{\prime \prime }+\left (2 \,{\mathrm e}^{\lambda x} a +b \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+a \left (b +\lambda \right ) {\mathrm e}^{\lambda x}+c \right ) y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.9 |
|
|
\[ {}y^{\prime \prime }+\left ({\mathrm e}^{\lambda x} a +2 b -\lambda \right ) y^{\prime }+\left (c \,{\mathrm e}^{2 \lambda x}+a b \,{\mathrm e}^{\lambda x}+b^{2}-b \lambda \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.588 |
|
|
\[ {}y^{\prime \prime }+\left (a \,{\mathrm e}^{x}+b \right ) y^{\prime }+\left (c \left (a -c \right ) {\mathrm e}^{2 x}+\left (a k +b c -2 c k +c \right ) {\mathrm e}^{x}+k \left (b -k \right )\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.474 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }+y x = x \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.967 |
|
|
\[ {}x y^{\prime \prime }-\left (1+2 x \right ) y^{\prime }+\left (1+x \right ) y = x^{2}-x -1 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.361 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.814 |
|
|
\[ {}\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = \left (1-x \right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.904 |
|
|
\[ {}\sin \left (x \right ) y^{\prime \prime }+2 \cos \left (x \right ) y^{\prime }+3 \sin \left (x \right ) y = {\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.505 |
|
|
\[ {}y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime }-\left (a^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.397 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }-y x = 2 \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.672 |
|
|
\[ {}y^{\prime \prime }+\left (2 \,{\mathrm e}^{x}-1\right ) y^{\prime }+{\mathrm e}^{2 x} y = {\mathrm e}^{4 x} \] |
1 |
1 |
1 |
second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.796 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x^{2}+1\right ) y^{\prime }-8 x^{3} y = 4 x^{3} {\mathrm e}^{-x^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.665 |
|
|
\[ {}x y^{\prime \prime }-\left (x +3\right ) y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[_Laguerre] |
✓ |
✓ |
1.316 |
|
|
\[ {}\left (x -3\right ) y^{\prime \prime }-\left (4 x -9\right ) y^{\prime }+\left (3 x -6\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.796 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (-x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.485 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.771 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x -1\right ) y^{\prime }+\left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.462 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (x^{2}+6\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.617 |
|
|
\[ {}\left (2 x^{3}-1\right ) y^{\prime \prime }-6 x^{2} y^{\prime }+6 y x = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.741 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x \left (1+x \right ) y^{\prime }+2 \left (1+x \right ) y = x^{3} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.819 |
|
|
\[ {}x^{5} y^{\prime \prime }+\left (2 x^{4}-x \right ) y^{\prime }-\left (2 x^{3}-1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.532 |
|
|
\[ {}\sin \left (x \right )^{2} y^{\prime \prime }-2 y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.667 |
|
|
\[ {}x^{\prime \prime }-t x^{\prime }+x = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[_Hermite] |
✓ |
✓ |
0.599 |
|
|
\[ {}x^{\prime \prime }-\frac {\left (2+t \right ) x^{\prime }}{t}+\frac {\left (2+t \right ) x}{t^{2}} = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.397 |
|
|
\[ {}t^{2} x^{\prime \prime }+t x^{\prime }+\left (t^{2}-\frac {1}{4}\right ) x = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.362 |
|
|
\[ {}y^{2}+3+\left (2 y x -4\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.123 |
|
|
\[ {}y^{\prime } = -8 x y^{2}+4 x \left (1+4 x \right ) y-8 x^{3}-4 x^{2}+1 \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
1.658 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[_Gegenbauer] |
✓ |
✓ |
0.376 |
|
|
\[ {}\left (x^{2}-x +1\right ) y^{\prime \prime }-\left (x^{2}+x \right ) y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.563 |
|
|
\[ {}\left (1+2 x \right ) y^{\prime \prime }-4 \left (1+x \right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.375 |
|
|
\[ {}\left (x^{3}-x^{2}\right ) y^{\prime \prime }-\left (x^{3}+2 x^{2}-2 x \right ) y^{\prime }+\left (2 x^{2}+2 x -2\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.944 |
|
|
\[ {}\left (x^{2}+2 x \right ) y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+2 y = \left (2+x \right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.174 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (2+x \right ) y^{\prime }+\left (2+x \right ) y = x^{3} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.052 |
|
|
\[ {}x \left (x -2\right ) y^{\prime \prime }-\left (x^{2}-2\right ) y^{\prime }+2 \left (-1+x \right ) y = 3 x^{2} \left (x -2\right )^{2} {\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.849 |
|
|
\[ {}\sin \left (x \right )^{2} y^{\prime \prime }-2 \sin \left (x \right ) \cos \left (x \right ) y^{\prime }+\left (\cos \left (x \right )^{2}+1\right ) y = \sin \left (x \right )^{3} \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.048 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+\left (2+3 x \right ) y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.649 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+\left (-2+3 x \right ) y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.677 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-\left (-2+3 x \right ) y^{\prime }+2 y x = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.77 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }-2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.095 |
|
|
\[ {}\left (2 x^{2}-3\right ) y^{\prime \prime }-2 x y^{\prime }+y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.066 |
|
|
\[ {}3 x y^{\prime \prime }-\left (x -2\right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.998 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+y x = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Lienard] |
✓ |
✓ |
0.719 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.776 |
|
|
\[ {}x y^{\prime \prime }-\left (x^{2}+2\right ) y^{\prime }+y x = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Lienard] |
✓ |
✓ |
0.773 |
|
|
\[ {}x^{2} y^{\prime \prime }+x^{2} y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.791 |
|
|
\[ {}\left (2 x^{2}-x \right ) y^{\prime \prime }+\left (2 x -2\right ) y^{\prime }+\left (-2 x^{2}+3 x -2\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.011 |
|
|
\[ {}t^{2} y^{\prime \prime }-\left (t^{2}+2 t \right ) y^{\prime }+\left (2+t \right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.644 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.356 |
|
|
\[ {}\left (-t^{2}+t \right ) x^{\prime \prime }+\left (-t^{2}+2\right ) x^{\prime }+\left (-t +2\right ) x = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.422 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[_Hermite] |
✓ |
✓ |
0.251 |
|
|
\[ {}\tan \left (t \right ) x^{\prime \prime }-3 x^{\prime }+\left (\tan \left (t \right )+3 \cot \left (t \right )\right ) x = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.474 |
|
|
\[ {}\left (\tan \left (x \right )^{2}-1\right ) y^{\prime \prime }-4 \tan \left (x \right )^{3} y^{\prime }+2 y \sec \left (x \right )^{4} = \left (\tan \left (x \right )^{2}-1\right ) \left (1-2 \sin \left (x \right )^{2}\right ) \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.492 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Hermite] |
✓ |
✓ |
0.388 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }-4 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.417 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+4 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.522 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-6 y = 1 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.538 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.945 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.369 |
|
|
\[ {}x y^{\prime \prime }+x^{2} y^{\prime }+2 y x = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.46 |
|
|
\[ {}y y^{\prime \prime } \sin \left (x \right )+\left (\sin \left (x \right ) y^{\prime }+y \cos \left (x \right )\right ) y^{\prime } = \cos \left (x \right ) \] |
1 |
1 |
2 |
second_order_integrable_as_is |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.618 |
|
|
\[ {}y^{\prime \prime }+\frac {2 x y^{\prime }}{2 x -1}-\frac {4 x y}{\left (2 x -1\right )^{2}} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.691 |
|
|
\[ {}\left (x^{2}+2 x \right ) y^{\prime \prime }+\left (x^{2}+x +10\right ) y^{\prime } = \left (25-6 x \right ) y \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.045 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{1+x}-\frac {\left (2+x \right ) y}{x^{2} \left (1+x \right )} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.211 |
|
|
\[ {}\left (x^{2}-x \right ) y^{\prime \prime }+\left (2 x^{2}+4 x -3\right ) y^{\prime }+8 y x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.879 |
|
|
\[ {}y^{\prime \prime }+\left (5+2 x \right ) y^{\prime }+\left (4 x +8\right ) y = {\mathrm e}^{-2 x} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.538 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.753 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+y x = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[_Lienard] |
✓ |
✓ |
0.353 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x}+\left (1-\frac {1}{4 x^{2}}\right ) y = x \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.238 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x -\frac {1}{2}\right ) y^{\prime }+\frac {y}{2} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.46 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.0 |
|
|
\[ {}x \left (1-x \right ) y^{\prime \prime }+\left (1-5 x \right ) y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[_Jacobi] |
✓ |
✓ |
0.875 |
|
|
\[ {}\left (x^{2}-1\right )^{2} y^{\prime \prime }+\left (1+x \right ) y^{\prime }-y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.156 |
|
|
\[ {}x y^{\prime \prime }+4 y^{\prime }-y x = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.755 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.949 |
|
|
\[ {}y^{\prime \prime }-\left (4+\frac {2}{x}\right ) y^{\prime }+\left (4+\frac {4}{x}\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.371 |
|
|
\[ {}\left (1+x \right ) y^{\prime \prime }+x y^{\prime }-y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.366 |
|
|
\[ {}\sin \left (x \right )^{2} y^{\prime \prime }-2 \sin \left (x \right ) \cos \left (x \right ) y^{\prime }+\left (\cos \left (x \right )^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.672 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.332 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.358 |
|
|
\[ {}\left (1+x \right ) y^{\prime \prime }+x y^{\prime }-y = \left (1+x \right )^{2} \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.282 |
|
|
\[ {}\left (1+x \right ) y^{\prime \prime }+x y^{\prime }-y = \left (1+x \right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.867 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+6 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.575 |
|
|
\[ {}y^{\prime \prime }+\left (2+x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.504 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-2 y x = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.494 |
|
|
\[ {}3 \left (x -2\right )^{2} y^{\prime \prime }-4 \left (x -5\right ) y^{\prime }+2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.71 |
|
|
\[ {}\left (x^{2}+4\right )^{2} y^{\prime \prime }+y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.408 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.813 |
|
|
\[ {}\left (-9 x^{4}+x^{2}\right ) y^{\prime \prime }-6 x y^{\prime }+10 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.851 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+\frac {y}{1-x} = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.78 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+\left (-2 x^{3}+5 x \right ) y^{\prime }+\left (-x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.57 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (2 x^{2}+5 x \right ) y^{\prime }+\left (9+4 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.783 |
|
|
\[ {}\left (-3 x^{3}+3 x^{2}\right ) y^{\prime \prime }-\left (5 x^{2}+4 x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.94 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (x^{2}+x \right ) y^{\prime }+4 y x = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Laguerre] |
✓ |
✓ |
1.362 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (-x^{4}+x \right ) y^{\prime }+3 x^{3} y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.838 |
|
|
\[ {}\left (9 x^{3}+9 x^{2}\right ) y^{\prime \prime }+\left (27 x^{2}+9 x \right ) y^{\prime }+\left (8 x -1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.925 |
|
|
\[ {}y^{\prime \prime }+\frac {2 y^{\prime }}{2+x}+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.82 |
|
|
\[ {}\left (x -3\right )^{2} y^{\prime \prime }+\left (x^{2}-3 x \right ) y^{\prime }-3 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.984 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (-x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.296 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x^{2} y^{\prime }+\left (x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.829 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (2 x^{2}+5 x \right ) y^{\prime }+9 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.771 |
|
|
\[ {}x y^{\prime \prime }+4 y^{\prime }+\frac {12 y}{\left (2+x \right )^{2}} = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.13 |
|
|
\[ {}x y^{\prime \prime }+4 y^{\prime }+\frac {12 y}{\left (2+x \right )^{2}} = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.958 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (x^{2}+x \right ) y^{\prime }+4 y x = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Laguerre] |
✓ |
✓ |
1.295 |
|
|
\[ {}t^{2} y^{\prime \prime }+t y^{\prime }+\left (t^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.365 |
|
|
\[ {}4 t^{2} y^{\prime \prime }+4 t y^{\prime }+\left (36 t^{2}-1\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.386 |
|
|
\[ {}t y^{\prime \prime }+2 y^{\prime }+16 t y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.413 |
|
|
\[ {}t^{2} y^{\prime \prime }-4 t y^{\prime }+\left (t^{2}+6\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.362 |
|
|
\[ {}t^{2} y^{\prime \prime }-4 t y^{\prime }+\left (t^{2}+6\right ) y = t^{3}+2 t \] |
1 |
0 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.822 |
|
|
\[ {}t y^{\prime \prime }+2 y^{\prime }+t y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[_Lienard] |
✓ |
✓ |
0.319 |
|
|
\[ {}t y^{\prime \prime }+2 y^{\prime }+t y = -t \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.057 |
|
|
\[ {}4 t^{2} y^{\prime \prime }+4 t y^{\prime }+\left (16 t^{2}-1\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.907 |
|
|
\[ {}4 t^{2} y^{\prime \prime }+4 t y^{\prime }+\left (16 t^{2}-1\right ) y = 16 t^{\frac {3}{2}} \] |
1 |
0 |
0 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
0.896 |
|
|
\[ {}4 t^{2} y^{\prime \prime }+4 t y^{\prime }+\left (16 t^{2}-1\right ) y = 16 t^{\frac {3}{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.977 |
|
|
\[ {}\left (t^{2}+t \right ) y^{\prime \prime \prime }+\left (-t^{2}+2\right ) y^{\prime \prime }-\left (2+t \right ) y^{\prime } = -2-t \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.701 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+4 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Hermite] |
✓ |
✓ |
0.263 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.747 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+6 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.631 |
|
|
\[ {}2 x y^{\prime \prime }-5 y^{\prime }-3 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.951 |
|
|
\[ {}y^{\prime \prime }+\left (\frac {16}{3 x}-1\right ) y^{\prime }-\frac {16 y}{3 x^{2}} = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.957 |
|
|
\[ {}x \left (1-x \right ) y^{\prime \prime }+\left (-2 x +1\right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[_Jacobi] |
✓ |
✓ |
0.764 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.808 |
|
|
\[ {}t y^{\prime \prime }+2 y^{\prime }+t y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[_Lienard] |
✓ |
✓ |
0.348 |
|
|
\[ {}y^{\prime \prime }-2 t y^{\prime }+t^{2} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.77 |
|
|
\[ {}x \left (1-x \right ) y^{\prime \prime }+\left (1+2 x \right ) y^{\prime }+10 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[_Jacobi] |
✓ |
✓ |
0.897 |
|
|
\[ {}x \left (1+x \right ) y^{\prime \prime }+\left (-2 x +1\right ) y^{\prime }-10 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.887 |
|
|
\[ {}y^{\prime }-2 \,{\mathrm e}^{x} y = 2 \sqrt {{\mathrm e}^{x} y} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.282 |
|
|
\[ {}\left (1+2 x \right ) y^{\prime \prime }+\left (-2+4 x \right ) y^{\prime }-8 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.658 |
|
|
\[ {}\left (x^{2}-x \right ) y^{\prime \prime }+\left (2 x -3\right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[_Jacobi] |
✓ |
✓ |
1.023 |
|
|
\[ {}\left (2 x^{2}+3 x \right ) y^{\prime \prime }-6 \left (1+x \right ) y^{\prime }+6 y = 6 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.009 |
|
|
\[ {}x^{2} \left (-1+\ln \left (x \right )\right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.188 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = \left (-1+x \right )^{2} {\mathrm e}^{x} \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.581 |
|
|
\[ {}\left (x^{4}-x^{3}\right ) y^{\prime \prime }+\left (2 x^{3}-2 x^{2}-x \right ) y^{\prime }-y = \frac {\left (-1+x \right )^{2}}{x} \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.247 |
|
|
\[ {}x \left (-1+x \right ) y^{\prime \prime }-\left (2 x -1\right ) y^{\prime }+2 y = x^{2} \left (2 x -3\right ) \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.236 |
|
|
\[ {}\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = \left (-1+x \right )^{2} {\mathrm e}^{x} \] |
1 |
0 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.191 |
|
|
\[ {}2 x^{2} \left (2-\ln \left (x \right )\right ) y^{\prime \prime }+x \left (4-\ln \left (x \right )\right ) y^{\prime }-y = \frac {\left (2-\ln \left (x \right )\right )^{2}}{\sqrt {x}} \] |
1 |
0 |
1 |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
0.681 |
|
|
\[ {}y^{\prime \prime }+\frac {2 y^{\prime }}{x}-y = 4 \,{\mathrm e}^{x} \] |
1 |
0 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
3.046 |
|
|
\[ {}\left (x^{2}-2 x \right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }-2 \left (1-x \right ) y = 2 x -2 \] |
1 |
0 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.925 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+y = 1 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.802 |
|
|
\[ {}y^{\prime \prime }-\left (x^{2}+1\right ) y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.426 |
|
|
\[ {}9 x \left (1-x \right ) y^{\prime \prime }-12 y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[_Jacobi] |
✓ |
✓ |
1.044 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.631 |
|
|
|
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|
|
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|
|