|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}y^{\prime \prime }+y-\sin \left (a x \right ) \sin \left (b x \right ) = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.248 |
|
|
\[
{}y^{\prime \prime }-y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.651 |
|
|
\[
{}y^{\prime \prime }-2 y-4 x^{2} {\mathrm e}^{x^{2}} = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
0.600 |
|
|
\[
{}y^{\prime \prime }+a^{2} y-\cot \left (a x \right ) = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.290 |
|
|
\[
{}y^{\prime \prime }+l y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.769 |
|
|
\[
{}y^{\prime \prime }+\left (a x +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.222 |
|
|
\[
{}y^{\prime \prime }-\left (x^{2}+1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.367 |
|
|
\[
{}y^{\prime \prime }-\left (x^{2}+a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.656 |
|
|
\[
{}y^{\prime \prime }-\left (a^{2} x^{2}+a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.487 |
|
|
\[
{}y^{\prime \prime }-c \,x^{a} y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.306 |
|
|
\[
{}y^{\prime \prime }-\left (a^{2} x^{2 n}-1\right ) y = 0
\] |
[_Titchmarsh] |
✗ |
0.533 |
|
|
\[
{}y^{\prime \prime }+\left (a \,x^{2 c}+b \,x^{c -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.545 |
|
|
\[
{}y^{\prime \prime }+\left ({\mathrm e}^{2 x}-v^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.323 |
|
|
\[
{}y^{\prime \prime }+a \,{\mathrm e}^{b x} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.302 |
|
|
\[
{}y^{\prime \prime }-\left (4 a^{2} b^{2} x^{2} {\mathrm e}^{2 b \,x^{2}}-1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.608 |
|
|
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{2 x}+b \,{\mathrm e}^{x}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.583 |
|
|
\[
{}y^{\prime \prime }+\left (a \cosh \left (x \right )^{2}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.129 |
|
|
\[
{}y^{\prime \prime }+\left (a \cos \left (2 x \right )+b \right ) y = 0
\] |
[_ellipsoidal] |
✗ |
1.001 |
|
|
\[
{}y^{\prime \prime }+\left (a \cos \left (x \right )^{2}+b \right ) y = 0
\] |
[_ellipsoidal] |
✗ |
1.360 |
|
|
\[
{}y^{\prime \prime }-\left (1+2 \tan \left (x \right )^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.137 |
|
|
\[
{}y^{\prime \prime }-\left (\frac {m \left (m -1\right )}{\cos \left (x \right )^{2}}+\frac {n \left (n -1\right )}{\sin \left (x \right )^{2}}+a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
6.380 |
|
|
\[
{}y^{\prime \prime }-\left (n \left (n +1\right ) k^{2} \operatorname {JacobiSN}\left (x , k\right )^{2}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.534 |
|
|
\[
{}y^{\prime \prime }-\left (f \left (x \right )^{2}+f^{\prime }\left (x \right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.583 |
|
|
\[
{}y^{\prime \prime }+y^{\prime }+a \,{\mathrm e}^{-2 x} y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
0.988 |
|
|
\[
{}y^{\prime \prime }-y^{\prime }+{\mathrm e}^{2 x} y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
0.604 |
|
|
\[
{}y^{\prime \prime }+a y^{\prime }+b y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.402 |
|
|
\[
{}y^{\prime \prime }+a y^{\prime }+b y-f \left (x \right ) = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
6.101 |
|
|
\[
{}y^{\prime \prime }+a y^{\prime }-\left (b^{2} x^{2}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.931 |
|
|
\[
{}y^{\prime \prime }+2 a y^{\prime }+f \left (x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.639 |
|
|
\[
{}y^{\prime \prime }+y^{\prime } x +y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
0.766 |
|
|
\[
{}y^{\prime \prime }+y^{\prime } x -y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.696 |
|
|
\[
{}y^{\prime \prime }+y^{\prime } x +\left (n +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.845 |
|
|
\[
{}y^{\prime \prime }+y^{\prime } x -n y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.860 |
|
|
\[
{}y^{\prime \prime }-y^{\prime } x +2 y = 0
\] |
[_Hermite] |
✓ |
0.537 |
|
|
\[
{}y^{\prime \prime }-y^{\prime } x -a y = 0
\] |
[_Hermite] |
✗ |
0.789 |
|
|
\[
{}y^{\prime \prime }-y^{\prime } x +\left (x -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
0.385 |
|
|
\[
{}y^{\prime \prime }-2 y^{\prime } x +a y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.775 |
|
|
\[
{}y^{\prime \prime }+4 y^{\prime } x +\left (4 x^{2}+2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.525 |
|
|
\[
{}y^{\prime \prime }-4 y^{\prime } x +\left (3 x^{2}+2 n -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.847 |
|
|
\[
{}y^{\prime \prime }-4 y^{\prime } x +\left (4 x^{2}-1\right ) y-{\mathrm e}^{x} = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
0.810 |
|
|
\[
{}y^{\prime \prime }-4 y^{\prime } x +\left (4 x^{2}-2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.448 |
|
|
\[
{}y^{\prime \prime }-4 y^{\prime } x +\left (4 x^{2}-3\right ) y-{\mathrm e}^{x^{2}} = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
0.734 |
|
|
\[
{}y^{\prime \prime }+a x y^{\prime }+b y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.874 |
|
|
\[
{}y^{\prime \prime }+2 a x y^{\prime }+a^{2} x^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.314 |
|
|
\[
{}y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (c x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.134 |
|
|
\[
{}y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x +\operatorname {c1} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.995 |
|
|
\[
{}y^{\prime \prime }-x^{2} y^{\prime }+x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.817 |
|
|
\[
{}y^{\prime \prime }-x^{2} y^{\prime }-\left (x +1\right )^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.647 |
|
|
\[
{}y^{\prime \prime }-x^{2} \left (x +1\right ) y^{\prime }+x \left (x^{4}-2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.582 |
|
|
\[
{}y^{\prime \prime }+x^{4} y^{\prime }-x^{3} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.928 |
|
|
\[
{}y^{\prime \prime }+a \,x^{q -1} y^{\prime }+b \,x^{q -2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.929 |
|
|
\[
{}y^{\prime \prime }+\sqrt {x}\, y^{\prime }+\left (\frac {1}{4 \sqrt {x}}+\frac {x}{4}-9\right ) y-x \,{\mathrm e}^{-\frac {x^{{3}/{2}}}{3}} = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
0.959 |
|
|
\[
{}y^{\prime \prime }-\frac {y^{\prime }}{\sqrt {x}}+\frac {\left (x +\sqrt {x}-8\right ) y}{4 x^{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.544 |
|
|
\[
{}y^{\prime \prime }-\left (2 \,{\mathrm e}^{x}+1\right ) y^{\prime }+{\mathrm e}^{2 x} y-{\mathrm e}^{3 x} = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.305 |
|
|
\[
{}y^{\prime \prime }+a y^{\prime }+\tan \left (x \right )+b y = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
37.505 |
|
|
\[
{}y^{\prime \prime }+2 n y^{\prime } \cot \left (x \right )+\left (-a^{2}+n^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.063 |
|
|
\[
{}y^{\prime \prime }+y^{\prime } \tan \left (x \right )+\cos \left (x \right )^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.988 |
|
|
\[
{}y^{\prime \prime }+y^{\prime } \tan \left (x \right )-\cos \left (x \right )^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.025 |
|
|
\[
{}y^{\prime \prime }+\cot \left (x \right ) y^{\prime }+v \left (v +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.751 |
|
|
\[
{}y^{\prime \prime }-\cot \left (x \right ) y^{\prime }+y \sin \left (x \right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.107 |
|
|
\[
{}y^{\prime \prime }+a y^{\prime } \tan \left (x \right )+b y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.726 |
|
|
\[
{}y^{\prime \prime }+2 a y^{\prime } \cot \left (a x \right )+\left (-a^{2}+b^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.777 |
|
|
\[
{}y^{\prime \prime }+f \left (x \right ) y^{\prime }+\left (\frac {f \left (x \right )^{2}}{4}+\frac {f^{\prime }\left (x \right )}{2}+a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.972 |
|
|
\[
{}y^{\prime \prime }-\frac {a f^{\prime }\left (x \right ) y^{\prime }}{f \left (x \right )}+b f \left (x \right )^{2 a} y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.122 |
|
|
\[
{}y^{\prime \prime }-\left (\frac {f^{\prime }\left (x \right )}{f \left (x \right )}+2 a \right ) y^{\prime }+\left (\frac {a f^{\prime }\left (x \right )}{f \left (x \right )}+a^{2}-b^{2} f \left (x \right )^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.102 |
|
|
\[
{}y^{\prime \prime }-\left (\frac {2 f^{\prime }\left (x \right )}{f \left (x \right )}+\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}-\frac {g^{\prime }\left (x \right )}{g \left (x \right )}\right ) y^{\prime }+\left (\frac {f^{\prime }\left (x \right ) \left (\frac {2 f^{\prime }\left (x \right )}{f \left (x \right )}+\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}-\frac {g^{\prime }\left (x \right )}{g \left (x \right )}\right )}{f \left (x \right )}-\frac {f^{\prime \prime }\left (x \right )}{f \left (x \right )}-\frac {v^{2} {g^{\prime }\left (x \right )}^{2}}{g \left (x \right )^{2}}+{g^{\prime }\left (x \right )}^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.034 |
|
|
\[
{}y^{\prime \prime }-\left (\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}+\frac {\left (2 v -1\right ) g^{\prime }\left (x \right )}{g \left (x \right )}+\frac {2 h^{\prime }\left (x \right )}{h \left (x \right )}\right ) y^{\prime }+\left (\frac {h^{\prime }\left (x \right ) \left (\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}+\frac {\left (2 v -1\right ) g^{\prime }\left (x \right )}{g \left (x \right )}+\frac {2 h^{\prime }\left (x \right )}{h \left (x \right )}\right )}{h \left (x \right )}-\frac {h^{\prime \prime }\left (x \right )}{h \left (x \right )}+{g^{\prime }\left (x \right )}^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.999 |
|
|
\[
{}4 y^{\prime \prime }+9 x y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.312 |
|
|
\[
{}4 y^{\prime \prime }-\left (x^{2}+a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.691 |
|
|
\[
{}4 y^{\prime \prime }+4 y^{\prime } \tan \left (x \right )-\left (5 \tan \left (x \right )^{2}+2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.271 |
|
|
\[
{}a y^{\prime \prime }-\left (a b +c +x \right ) y^{\prime }+\left (b \left (x +c \right )+d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.348 |
|
|
\[
{}a^{2} y^{\prime \prime }+a \left (a^{2}-2 b \,{\mathrm e}^{-a x}\right ) y^{\prime }+b^{2} {\mathrm e}^{-2 a x} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.957 |
|
|
\[
{}x \left (y^{\prime \prime }+y\right )-\cos \left (x \right ) = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
0.862 |
|
|
\[
{}x y^{\prime \prime }+\left (x +a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.706 |
|
|
\[
{}x y^{\prime \prime }+y^{\prime } = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.769 |
|
|
\[
{}x y^{\prime \prime }+y^{\prime }+a y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.351 |
|
|
\[
{}x y^{\prime \prime }+y^{\prime }+l x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.540 |
|
|
\[
{}x y^{\prime \prime }+y^{\prime }+\left (x +a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.937 |
|
|
\[
{}x y^{\prime \prime }-y^{\prime }+a y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.354 |
|
|
\[
{}x y^{\prime \prime }-y^{\prime }-y a \,x^{3} = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.324 |
|
|
\[
{}x y^{\prime \prime }-y^{\prime }+x^{3} \left ({\mathrm e}^{x^{2}}-v^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.782 |
|
|
\[
{}x y^{\prime \prime }+2 y^{\prime }-x y-{\mathrm e}^{x} = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.232 |
|
|
\[
{}x y^{\prime \prime }+2 y^{\prime }+y a x = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.863 |
|
|
\[
{}x y^{\prime \prime }+2 y^{\prime }+a \,x^{2} y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.311 |
|
|
\[
{}x y^{\prime \prime }-2 y^{\prime }+a y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.367 |
|
|
\[
{}x y^{\prime \prime }+v y^{\prime }+a y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.458 |
|
|
\[
{}x y^{\prime \prime }+a y^{\prime }+b x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.638 |
|
|
\[
{}x y^{\prime \prime }+a y^{\prime }+b \,x^{\operatorname {a1}} y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.390 |
|
|
\[
{}x y^{\prime \prime }+\left (x +b \right ) y^{\prime }+a y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.179 |
|
|
\[
{}x y^{\prime \prime }+\left (x +a +b \right ) y^{\prime }+a y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.378 |
|
|
\[
{}x y^{\prime \prime }-y^{\prime } x -y-x \left (x +1\right ) {\mathrm e}^{x} = 0
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
1.079 |
|
|
\[
{}x y^{\prime \prime }-y^{\prime } x -a y = 0
\] |
[_Laguerre] |
✗ |
0.747 |
|
|
\[
{}x y^{\prime \prime }-\left (x +1\right ) y^{\prime }+y = 0
\] |
[_Laguerre] |
✓ |
0.607 |
|
|
\[
{}x y^{\prime \prime }-\left (x +1\right ) y^{\prime }-2 \left (x -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.513 |
|
|
\[
{}x y^{\prime \prime }+\left (b -x \right ) y^{\prime }-a y = 0
\] |
[_Laguerre] |
✗ |
1.219 |
|
|
\[
{}x y^{\prime \prime }-2 \left (x -1\right ) y^{\prime }-y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.813 |
|
|
\[
{}x y^{\prime \prime }-\left (3 x -2\right ) y^{\prime }-\left (2 x -3\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.971 |
|
|
\[
{}x y^{\prime \prime }+\left (a x +b +n \right ) y^{\prime }+n a y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.466 |
|
|
\[
{}x y^{\prime \prime }-\left (a +b \right ) \left (x +1\right ) y^{\prime }+a b x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.540 |
|
|
\[
{}x y^{\prime \prime }+\left (\left (a +b \right ) x +m +n \right ) y^{\prime }+\left (a b x +a n +b m \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.968 |
|