|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}y^{2}+{y^{\prime }}^{2} = 1 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.601 |
|
|
\[ {}\left (2 x y^{\prime }-y\right )^{2} = 8 x^{3} \] |
linear |
[_linear] |
✓ |
✓ |
0.988 |
|
|
\[ {}\left (x^{2}+1\right ) {y^{\prime }}^{2} = 1 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.446 |
|
|
\[ {}{y^{\prime }}^{3}-\left (y^{2}+2 x \right ) {y^{\prime }}^{2}+\left (x^{2}-y^{2}+2 x y^{2}\right ) y^{\prime }-\left (x^{2}-y^{2}\right ) y^{2} = 0 \] |
linear, quadrature |
[_quadrature] |
✓ |
✓ |
0.855 |
|
|
\[ {}2 x y^{\prime }-y+\ln \left (y^{\prime }\right ) = 0 \] |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
1.593 |
|
|
\[ {}4 x {y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.349 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }-x = 0 \] |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.381 |
|
|
\[ {}y^{\prime }+2 x y = x^{2}+y^{2} \] |
riccati, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
2.116 |
|
|
\[ {}y = -x y^{\prime }+x^{4} {y^{\prime }}^{2} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
5.934 |
|
|
\[ {}{y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.498 |
|
|
\[ {}x +y^{\prime } y \left (2 {y^{\prime }}^{2}+3\right ) = 0 \] |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
140.131 |
|
|
\[ {}a^{2} y {y^{\prime }}^{2}-2 x y^{\prime }+y = 0 \] |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.736 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }-x = 0 \] |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.397 |
|
|
\[ {}{y^{\prime }}^{3}-4 x y y^{\prime }+8 y^{2} = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
136.206 |
|
|
\[ {}\left (-y+x y^{\prime }\right )^{2} = {y^{\prime }}^{2}+1 \] |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
0.709 |
|
|
\[ {}4 \,{\mathrm e}^{2 y} {y^{\prime }}^{2}+2 x y^{\prime }-1 = 0 \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
4.438 |
|
|
\[ {}4 \,{\mathrm e}^{2 y} {y^{\prime }}^{2}+2 \,{\mathrm e}^{2 x} y^{\prime }-{\mathrm e}^{2 x} = 0 \] |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.338 |
|
|
\[ {}{\mathrm e}^{2 y} {y^{\prime }}^{3}+\left ({\mathrm e}^{2 x}+{\mathrm e}^{3 x}\right ) y^{\prime }-{\mathrm e}^{3 x} = 0 \] |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
204.26 |
|
|
\[ {}x y^{2} {y^{\prime }}^{2}-y^{3} y^{\prime }+x = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
10.509 |
|
|
\[ {}\left (x^{2}+y^{2}\right ) \left (1+y^{\prime }\right )^{2}-2 \left (x +y\right ) \left (1+y^{\prime }\right ) \left (x +y y^{\prime }\right )+\left (x +y y^{\prime }\right )^{2} = 0 \] |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
3.897 |
|
|
\[ {}y = 2 x y^{\prime }+y^{2} {y^{\prime }}^{3} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
133.768 |
|
|
\[ {}a^{2} y {y^{\prime }}^{2}-2 x y^{\prime }+y = 0 \] |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.736 |
|
|
\[ {}\left (x -y^{\prime }-y\right )^{2} = x^{2} \left (2 x y-x^{2} y^{\prime }\right ) \] |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
5.846 |
|
|
\[ {}y^{2} \left ({y^{\prime }}^{2}+1\right ) = a^{2} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
1.152 |
|
|
\[ {}y y^{\prime } = \left (-b +x \right ) {y^{\prime }}^{2}+a \] |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.422 |
|
|
\[ {}x^{3} {y^{\prime }}^{2}+x^{2} y y^{\prime }+1 = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
6.652 |
|
|
\[ {}3 x {y^{\prime }}^{2}-6 y y^{\prime }+x +2 y = 0 \] |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.42 |
|
|
\[ {}y = {y^{\prime }}^{2} \left (1+x \right ) \] |
dAlembert, first_order_nonlinear_p_but_separable |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
✓ |
1.473 |
|
|
\[ {}\left (-y+x y^{\prime }\right ) \left (x +y y^{\prime }\right ) = a^{2} y^{\prime } \] |
first_order_ode_lie_symmetry_calculated |
[_rational] |
✓ |
✓ |
72.443 |
|
|
\[ {}{y^{\prime }}^{2}+2 y^{\prime } y \cot \left (x \right ) = y^{2} \] |
separable |
[_separable] |
✓ |
✓ |
6.134 |
|
|
\[ {}\left (x^{2}+1\right ) {y^{\prime }}^{2}-2 x y y^{\prime }+y^{2}-1 = 0 \] |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
0.564 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-2 \left (x y+2 y^{\prime }\right ) y^{\prime }+y^{2} = 0 \] |
separable |
[_separable] |
✓ |
✓ |
0.957 |
|
|
\[ {}y = x y^{\prime }+\frac {y {y^{\prime }}^{2}}{x^{2}} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
8.293 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+y^{2} = x^{2} y^{2}+x^{4} \] |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
3.427 |
|
|
\[ {}y = x y^{\prime }+\frac {1}{y^{\prime }} \] |
clairaut |
[[_homogeneous, ‘class G‘], _rational, _Clairaut] |
✓ |
✓ |
0.363 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }-x = 0 \] |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.373 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-2 \left (x y-2\right ) y^{\prime }+y^{2} = 0 \] |
clairaut |
[[_homogeneous, ‘class G‘], _Clairaut] |
✓ |
✓ |
0.776 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-\left (-1+x \right )^{2} = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.37 |
|
|
\[ {}8 \left (1+y^{\prime }\right )^{3} = 27 \left (x +y\right ) \left (1-y^{\prime }\right )^{3} \] |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
473.104 |
|
|
\[ {}4 {y^{\prime }}^{2} = 9 x \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.282 |
|
|
\[ {}y \left (3-4 y\right )^{2} {y^{\prime }}^{2} = 4-4 y \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.886 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.229 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+25 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.593 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime } = 0 \] |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.228 |
|
|
\[ {}y^{\prime \prime \prime }-2 y^{\prime \prime }-y^{\prime }+2 y = 0 \] |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.223 |
|
|
\[ {}4 y^{\prime \prime \prime }-3 y^{\prime }+y = 0 \] |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.281 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime \prime }-y^{\prime }+y = 0 \] |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.253 |
|
|
\[ {}y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }-2 y^{\prime }-y = 0 \] |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.119 |
|
|
\[ {}y^{\prime \prime \prime }-6 y^{\prime \prime }+9 y^{\prime } = 0 \] |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.272 |
|
|
\[ {}y^{\prime \prime \prime \prime }+2 y^{\prime \prime }+y = 0 \] |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.122 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime \prime }+y^{\prime } = 0 \] |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.531 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime \prime }-2 y^{\prime } = {\mathrm e}^{-x} \] |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.562 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = {\mathrm e}^{{\mathrm e}^{x}} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.498 |
|
|
\[ {}y^{\prime \prime \prime }+3 y^{\prime \prime }+3 y^{\prime }+y = 2 \,{\mathrm e}^{-x}-x^{2} {\mathrm e}^{-x} \] |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.234 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = \frac {{\mathrm e}^{x}}{\left (1-x \right )^{2}} \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.519 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = {\mathrm e}^{x} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.42 |
|
|
\[ {}y^{\prime \prime \prime }-3 y^{\prime \prime }-y^{\prime }+3 y = x^{2} \] |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.583 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.739 |
|
|
\[ {}y^{\prime \prime \prime }-4 y^{\prime \prime }+5 y^{\prime }-2 y = x \] |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.612 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.461 |
|
|
\[ {}y^{\prime \prime }+y = \tan \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.758 |
|
|
\[ {}y^{\prime \prime }+4 y = x^{2}+\cos \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.291 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 2 \,{\mathrm e}^{2 x} x -\sin \left (x \right )^{2} \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.616 |
|
|
\[ {}y^{\prime \prime }+y = 2 \,{\mathrm e}^{x}+x^{3}-x \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.648 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 3 \,{\mathrm e}^{2 x}-\cos \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.894 |
|
|
\[ {}y^{\prime \prime \prime }-y = x^{2} \] |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.954 |
|
|
\[ {}y^{\prime \prime \prime }-2 y^{\prime \prime }-3 y^{\prime } = 3 x^{2}+\sin \left (x \right ) \] |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.908 |
|
|
\[ {}y^{\prime \prime \prime \prime }-2 y^{\prime \prime }+y = {\mathrm e}^{x}+4 \] |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
0.211 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime } = {\mathrm e}^{2 x}+1 \] |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.929 |
|
|
\[ {}y^{\prime \prime \prime \prime }+2 y^{\prime \prime }+y = \cos \left (x \right ) \] |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.004 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }+x y^{\prime }-y = x \ln \left (x \right ) \] |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.353 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }+2 x^{2} y^{\prime \prime }+2 y = 10 x +\frac {10}{x} \] |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_3rd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.912 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+y = \frac {1}{\left (1-x \right )^{2}} \] |
kovacic, second_order_euler_ode, exact linear second order ode, second_order_integrable_as_is, 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_2 |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.465 |
|
|
\[ {}\left (1+x \right )^{2} y^{\prime \prime }-\left (1+x \right ) y^{\prime }+6 y = x \] |
kovacic, second_order_change_of_variable_on_x_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
6.039 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = \cos \left (x \right )-{\mathrm e}^{2 x} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.721 |
|
|
\[ {}y^{\prime \prime \prime \prime }-y = {\mathrm e}^{x} \cos \left (x \right ) \] |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.065 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 2 x^{3}-x \,{\mathrm e}^{3 x} \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.601 |
|
|
\[ {}y^{\prime \prime \prime }-4 y^{\prime } = x^{2}-3 \,{\mathrm e}^{2 x} \] |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.587 |
|
|
\[ {}y^{\prime \prime \prime \prime }-2 y^{\prime \prime }+y = \cos \left (x \right ) \] |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.226 |
|
|
\[ {}x^{4} y^{\prime \prime \prime \prime }+6 x^{3} y^{\prime \prime \prime }+9 x^{2} y^{\prime \prime }+3 x y^{\prime }+y = \left (1+\ln \left (x \right )\right )^{2} \] |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.239 |
|
|
\[ {}y^{\prime \prime \prime }+2 y^{\prime \prime }+y^{\prime } = x^{2}-x \] |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.198 |
|
|
\[ {}y^{\prime \prime }+4 y = \sin \left (x \right )^{2} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.099 |
|
|
\[ {}y^{\prime \prime }+4 y = \sec \left (x \right )^{2} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.996 |
|
|
\[ {}y^{\prime \prime \prime \prime }-y^{\prime \prime \prime }-3 y^{\prime \prime }+5 y^{\prime }-2 y = {\mathrm e}^{3 x} \] |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
0.212 |
|
|
\[ {}y^{\prime \prime }+y = \cos \left (x \right ) x \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.797 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }+2 x^{2} y^{\prime \prime }-x y^{\prime }+y = \frac {1}{x} \] |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_3rd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.335 |
|
|
\[ {}y^{\prime \prime \prime }-y = x \,{\mathrm e}^{x}+\cos \left (x \right )^{2} \] |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
8.678 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }+x y = x \] |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.58 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +1\right ) y^{\prime }+\left (1+x \right ) y = x^{2}-x -1 \] |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.702 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.725 |
|
|
\[ {}\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = \left (1-x \right )^{2} \] |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.987 |
|
|
\[ {}\sin \left (x \right ) y^{\prime \prime }+2 \cos \left (x \right ) y^{\prime }+3 \sin \left (x \right ) y = {\mathrm e}^{x} \] |
kovacic, second_order_change_of_variable_on_y_method_1, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.663 |
|
|
\[ {}y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime }-\left (a^{2}+1\right ) y = 0 \] |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.949 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x^{3} y^{\prime }+\left (x^{2}+1\right ) y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.31 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }-x y = 2 \,{\mathrm e}^{x} \] |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.154 |
|
|
\[ {}y^{\prime \prime }+\left (2 \,{\mathrm e}^{x}-1\right ) y^{\prime }+{\mathrm e}^{2 x} y = {\mathrm e}^{4 x} \] |
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]] |
✓ |
✓ |
7.931 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+4 y = 0 \] |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.29 |
|
|
\[ {}y^{\prime \prime }+\tan \left (x \right ) y^{\prime }+\cos \left (x \right )^{2} y = 0 \] |
second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
7.284 |
|
|
\[ {}x^{6} y^{\prime \prime }+3 x^{5} y^{\prime }+y = \frac {1}{x^{2}} \] |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.721 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x^{2}+1\right ) y^{\prime }-8 x^{3} y = 4 x^{3} {\mathrm e}^{-x^{2}} \] |
kovacic, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.59 |
|
|
|
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