|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}y^{\prime \prime }-y^{\prime } \cot \left (x \right )+y \sin \left (x \right )^{2} = 0 \] |
second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
10.019 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime } \tan \left (x \right )+b y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.436 |
|
|
\[ {}y^{\prime \prime }+2 a y^{\prime } \cot \left (x a \right )+\left (-a^{2}+b^{2}\right ) y = 0 \] |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.75 |
|
|
\[ {}y^{\prime \prime }+a p^{\prime \prime }\left (x \right ) y^{\prime }+\left (a +b p \left (x \right )-4 n a p \left (x \right )^{2}\right ) y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.259 |
|
|
\[ {}y^{\prime \prime }+\frac {\left (11 \operatorname {WeierstrassP}\left (x , a , b\right ) \operatorname {WeierstrassPPrime}\left (x , a , b\right )-6 \operatorname {WeierstrassP}\left (x , a , b\right )^{2}+\frac {a}{2}\right ) y^{\prime }}{\operatorname {WeierstrassPPrime}\left (x , a , b\right )+\operatorname {WeierstrassP}\left (x , a , b\right )^{2}}+\frac {\left (\operatorname {WeierstrassPPrime}\left (x , a , b\right )^{2}-\operatorname {WeierstrassP}\left (x , a , b\right )^{2} \operatorname {WeierstrassPPrime}\left (x , a , b\right )-\operatorname {WeierstrassP}\left (x , a , b\right ) \left (6 \operatorname {WeierstrassP}\left (x , a , b\right )^{2}-\frac {a}{2}\right )\right ) y}{\operatorname {WeierstrassPPrime}\left (x , a , b\right )+\operatorname {WeierstrassP}\left (x , a , b\right )^{2}} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
86.672 |
|
|
\[ {}y^{\prime \prime }+f \left (x \right ) y^{\prime }+g \left (x \right ) y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.28 |
|
|
\[ {}y^{\prime \prime }+f \left (x \right ) y^{\prime }+\left (f^{\prime }\left (x \right )+a \right ) y-g \left (x \right ) = 0 \] |
unknown |
[[_2nd_order, _linear, _nonhomogeneous]] |
❇ |
N/A |
0.244 |
|
|
\[ {}y^{\prime \prime }+\left (a f \left (x \right )+b \right ) y^{\prime }+\left (c f \left (x \right )+d \right ) y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.681 |
|
|
\[ {}y^{\prime \prime }+f \left (x \right ) y^{\prime }+\left (\frac {f \left (x \right )^{2}}{4}+\frac {f^{\prime }\left (x \right )}{2}+a \right ) y = 0 \] |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.575 |
|
|
\[ {}y^{\prime \prime }-\frac {a f^{\prime }\left (x \right ) y^{\prime }}{f \left (x \right )}+b f \left (x \right )^{2 a} y = 0 \] |
second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.908 |
|
|
\[ {}y^{\prime \prime }-\left (\frac {f^{\prime }\left (x \right )}{f \left (x \right )}+2 a \right ) y^{\prime }+\left (\frac {a f^{\prime }\left (x \right )}{f \left (x \right )}+a^{2}-b^{2} f \left (x \right )^{2}\right ) y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.642 |
|
|
\[ {}y^{\prime \prime }+\frac {f \left (x \right ) f^{\prime \prime \prime }\left (x \right ) y^{\prime }}{f \left (x \right )^{2}+b^{2}}-\frac {a^{2} {f^{\prime }\left (x \right )}^{2} y}{f \left (x \right )^{2}+b^{2}} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.052 |
|
|
\[ {}y^{\prime \prime }-\left (\frac {2 f^{\prime }\left (x \right )}{f \left (x \right )}+\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}-\frac {g^{\prime }\left (x \right )}{g \left (x \right )}\right ) y^{\prime }+\left (\frac {f^{\prime }\left (x \right ) \left (\frac {2 f^{\prime }\left (x \right )}{f \left (x \right )}+\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}-\frac {g^{\prime }\left (x \right )}{g \left (x \right )}\right )}{f \left (x \right )}-\frac {f^{\prime \prime }\left (x \right )}{f \left (x \right )}-\frac {v^{2} {g^{\prime }\left (x \right )}^{2}}{g \left (x \right )^{2}}+{g^{\prime }\left (x \right )}^{2}\right ) y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.604 |
|
|
\[ {}y^{\prime \prime }-\left (\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}+\frac {\left (2 v -1\right ) g^{\prime }\left (x \right )}{g \left (x \right )}+\frac {2 h^{\prime }\left (x \right )}{h \left (x \right )}\right ) y^{\prime }+\left (\frac {h^{\prime }\left (x \right ) \left (\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}+\frac {\left (2 v -1\right ) g^{\prime }\left (x \right )}{g \left (x \right )}+\frac {2 h^{\prime }\left (x \right )}{h \left (x \right )}\right )}{h \left (x \right )}-\frac {h^{\prime \prime }\left (x \right )}{h \left (x \right )}+{g^{\prime }\left (x \right )}^{2}\right ) y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.644 |
|
|
\[ {}4 y^{\prime \prime }+9 x y = 0 \] |
second_order_airy, second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.422 |
|
|
\[ {}4 y^{\prime \prime }-\left (x^{2}+a \right ) y = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.875 |
|
|
\[ {}4 y^{\prime \prime }+4 y^{\prime } \tan \left (x \right )-\left (5 \tan \left (x \right )^{2}+2\right ) y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.287 |
|
|
\[ {}a y^{\prime \prime }-\left (a b +c +x \right ) y^{\prime }+\left (b \left (x +c \right )+d \right ) y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.168 |
|
|
\[ {}a^{2} y^{\prime \prime }+a \left (a^{2}-2 b \,{\mathrm e}^{-x a}\right ) y^{\prime }+b^{2} {\mathrm e}^{-2 x a} y = 0 \] |
second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.059 |
|
|
\[ {}x \left (y^{\prime \prime }+y\right )-\cos \left (x \right ) = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.277 |
|
|
\[ {}x y^{\prime \prime }+\left (x +a \right ) y = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.911 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime } = 0 \] |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.478 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime }+a y = 0 \] |
second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.458 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime }+l x y = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.707 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime }+\left (x +a \right ) y = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.947 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime }+a y = 0 \] |
second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.494 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime }-y a \,x^{3} = 0 \] |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.975 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime }+x^{3} \left ({\mathrm e}^{x^{2}}-v^{2}\right ) y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.431 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }-x y-{\mathrm e}^{x} = 0 \] |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.601 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+a x y = 0 \] |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.835 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+a \,x^{2} y = 0 \] |
second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.593 |
|
|
\[ {}x y^{\prime \prime }-2 y^{\prime }+a y = 0 \] |
second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.497 |
|
|
\[ {}x y^{\prime \prime }+v y^{\prime }+a y = 0 \] |
second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.578 |
|
|
\[ {}x y^{\prime \prime }+a y^{\prime }+b x y = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.937 |
|
|
\[ {}x y^{\prime \prime }+a y^{\prime }+b \,x^{\operatorname {a1}} y = 0 \] |
second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.444 |
|
|
\[ {}x y^{\prime \prime }+\left (x +b \right ) y^{\prime }+a y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.125 |
|
|
\[ {}x y^{\prime \prime }+\left (x +a +b \right ) y^{\prime }+a y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.261 |
|
|
\[ {}x y^{\prime \prime }-x y^{\prime }-y-x \left (1+x \right ) {\mathrm e}^{x} = 0 \] |
kovacic, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.437 |
|
|
\[ {}x y^{\prime \prime }-x y^{\prime }-a y = 0 \] |
unknown |
[_Laguerre] |
✗ |
N/A |
0.635 |
|
|
\[ {}x y^{\prime \prime }-\left (1+x \right ) y^{\prime }+y = 0 \] |
kovacic, second_order_ode_non_constant_coeff_transformation_on_B |
[_Laguerre] |
✓ |
✓ |
1.018 |
|
|
\[ {}x y^{\prime \prime }-\left (1+x \right ) y^{\prime }-2 \left (-1+x \right ) y = 0 \] |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.914 |
|
|
\[ {}x y^{\prime \prime }+\left (-x +b \right ) y^{\prime }-a y = 0 \] |
unknown |
[_Laguerre] |
✗ |
N/A |
1.073 |
|
|
\[ {}x y^{\prime \prime }-2 \left (-1+x \right ) y^{\prime }-y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.741 |
|
|
\[ {}x y^{\prime \prime }-\left (3 x -2\right ) y^{\prime }-\left (2 x -3\right ) y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.956 |
|
|
\[ {}x y^{\prime \prime }+\left (x a +b +n \right ) y^{\prime }+n a y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.461 |
|
|
\[ {}x y^{\prime \prime }-\left (a +b \right ) \left (1+x \right ) y^{\prime }+a b x y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.168 |
|
|
\[ {}x y^{\prime \prime }+\left (x \left (a +b \right )+m +n \right ) y^{\prime }+\left (a b x +a n +b m \right ) y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.511 |
|
|
\[ {}x y^{\prime \prime }-2 \left (x a +b \right ) y^{\prime }+\left (x \,a^{2}+2 a b \right ) y = 0 \] |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.359 |
|
|
\[ {}x y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+\left (c x +d \right ) y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.961 |
|
|
\[ {}x y^{\prime \prime }-\left (x^{2}-x \right ) y^{\prime }+\left (-1+x \right ) y = 0 \] |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.277 |
|
|
\[ {}x y^{\prime \prime }-\left (x^{2}-x -2\right ) y^{\prime }-x \left (x +3\right ) y = 0 \] |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.05 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x^{2} a +1\right ) y^{\prime }+b \,x^{3} y = 0 \] |
kovacic, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.61 |
|
|
\[ {}x y^{\prime \prime }-2 \left (x^{2}-a \right ) y^{\prime }+2 n x y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.467 |
|
|
\[ {}x y^{\prime \prime }+\left (4 x^{2}-1\right ) y^{\prime }-4 x^{3} y-4 x^{5} = 0 \] |
kovacic, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.376 |
|
|
\[ {}x y^{\prime \prime }+\left (2 a \,x^{3}-1\right ) y^{\prime }+\left (a^{2} x^{3}+a \right ) x^{2} y = 0 \] |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.217 |
|
|
\[ {}x y^{\prime \prime }+\left (2 a x \ln \left (x \right )+1\right ) y^{\prime }+\left (a^{2} x \ln \left (x \right )^{2}+a \ln \left (x \right )+a \right ) y = 0 \] |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.072 |
|
|
\[ {}x y^{\prime \prime }+\left (f \left (x \right ) x +2\right ) y^{\prime }+f \left (x \right ) y = 0 \] |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.907 |
|
|
\[ {}\left (x -3\right ) y^{\prime \prime }-\left (4 x -9\right ) y^{\prime }+\left (3 x -6\right ) y = 0 \] |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.245 |
|
|
\[ {}2 x y^{\prime \prime }+y^{\prime }+a y = 0 \] |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.266 |
|
|
\[ {}2 x y^{\prime \prime }-\left (-1+x \right ) y^{\prime }+a y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.946 |
|
|
\[ {}2 x y^{\prime \prime }-\left (2 x -1\right ) y^{\prime }+a y = 0 \] |
unknown |
[_Laguerre] |
✗ |
N/A |
0.937 |
|
|
\[ {}\left (2 x -1\right ) y^{\prime \prime }-\left (3 x -4\right ) y^{\prime }+\left (x -3\right ) y = 0 \] |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.496 |
|
|
\[ {}4 x y^{\prime \prime }-\left (x +a \right ) y = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.889 |
|
|
\[ {}4 x y^{\prime \prime }+2 y^{\prime }-y = 0 \] |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.116 |
|
|
\[ {}4 x y^{\prime \prime }+4 y^{\prime }-\left (2+x \right ) y = 0 \] |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.951 |
|
|
\[ {}4 x y^{\prime \prime }+4 y-\left (2+x \right ) y+l y = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.974 |
|
|
\[ {}4 x y^{\prime \prime }+4 m y^{\prime }-\left (x -2 m -4 n \right ) y = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.152 |
|
|
\[ {}16 x y^{\prime \prime }+8 y^{\prime }-\left (x +a \right ) y = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.112 |
|
|
\[ {}a x y^{\prime \prime }+b y^{\prime }+c y = 0 \] |
second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.673 |
|
|
\[ {}a x y^{\prime \prime }+\left (b x +3 a \right ) y^{\prime }+3 b y = 0 \] |
second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.4 |
|
|
\[ {}5 \left (x a +b \right ) y^{\prime \prime }+8 a y^{\prime }+c \left (x a +b \right )^{\frac {1}{5}} y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.859 |
|
|
\[ {}2 a x y^{\prime \prime }+\left (b x +a \right ) y^{\prime }+c y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.209 |
|
|
\[ {}2 a x y^{\prime \prime }+\left (b x +3 a \right ) y^{\prime }+c y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.21 |
|
|
\[ {}\left (\operatorname {a2} x +\operatorname {b2} \right ) y^{\prime \prime }+\left (\operatorname {a1} x +\operatorname {b1} \right ) y^{\prime }+\left (\operatorname {a0} x +\operatorname {b0} \right ) y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
9.317 |
|
|
\[ {}x^{2} y^{\prime \prime }-6 y = 0 \] |
kovacic, second_order_euler_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.43 |
|
|
\[ {}x^{2} y^{\prime \prime }-12 y = 0 \] |
kovacic, second_order_euler_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.429 |
|
|
\[ {}x^{2} y^{\prime \prime }+a y = 0 \] |
kovacic, second_order_euler_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.625 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x a +b \right ) y = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.53 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2}-2\right ) y = 0 \] |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.99 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (x^{2} a +2\right ) y = 0 \] |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.076 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (a^{2} x^{2}-6\right ) y = 0 \] |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.29 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2} a -v \left (v -1\right )\right ) y = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.909 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2} a +b x +c \right ) y = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
5.443 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (a \,x^{k}-b \left (b -1\right )\right ) y = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.313 |
|
|
\[ {}x^{2} y^{\prime \prime }+\frac {y}{\ln \left (x \right )}-x \,{\mathrm e}^{x} \left (2+x \ln \left (x \right )\right ) = 0 \] |
unknown |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
0.433 |
|
|
\[ {}x^{2} y^{\prime \prime }+a y^{\prime }-x y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.746 |
|
|
\[ {}x^{2} y^{\prime \prime }+a y^{\prime }-\left (b^{2} x^{2}+a b \right ) y = 0 \] |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.299 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y-x^{2} a = 0 \] |
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, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.91 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+a y = 0 \] |
kovacic, second_order_euler_ode, 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 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.013 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-\left (x +a \right ) y = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.573 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (-v^{2}+x^{2}\right ) y = 0 \] |
second_order_bessel_ode |
[_Bessel] |
✓ |
✓ |
0.852 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (-v^{2}+x^{2}\right ) y-f \left (x \right ) = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.898 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (l \,x^{2}-v^{2}\right ) y = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.954 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x +a \right ) y^{\prime }-y = 0 \] |
exact linear second order ode, second_order_integrable_as_is, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.319 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y-3 x^{3} = 0 \] |
kovacic, second_order_euler_ode, 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, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.866 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+\left (a \,x^{m}+b \right ) y = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.355 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime } = 0 \] |
kovacic, second_order_euler_ode, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.836 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }+\left (x a -b^{2}\right ) y = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.649 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }+\left (x^{2} a +b \right ) y = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.925 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }+\left (l \,x^{2}+x a -n \left (n +1\right )\right ) y = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
29.175 |
|
|
|
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