|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}x y^{\prime \prime }-2 \left (a x +b \right ) y^{\prime }+\left (a^{2} x +2 a b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.048 |
|
|
\[
{}x y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (c x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.940 |
|
|
\[
{}x y^{\prime \prime }-\left (x^{2}-x \right ) y^{\prime }+\left (-1+x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.252 |
|
|
\[
{}x y^{\prime \prime }-\left (x^{2}-x -2\right ) y^{\prime }-x \left (x +3\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.166 |
|
|
\[
{}x y^{\prime \prime }-\left (2 a \,x^{2}+1\right ) y^{\prime }+b \,x^{3} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.369 |
|
|
\[
{}x y^{\prime \prime }-2 \left (x^{2}-a \right ) y^{\prime }+2 n x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.155 |
|
|
\[
{}x y^{\prime \prime }+\left (4 x^{2}-1\right ) y^{\prime }-4 x^{3} y-4 x^{5} = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
2.000 |
|
|
\[
{}x y^{\prime \prime }+\left (2 a \,x^{3}-1\right ) y^{\prime }+\left (a^{2} x^{3}+a \right ) x^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.911 |
|
|
\[
{}x y^{\prime \prime }+\left (2 a x \ln \left (x \right )+1\right ) y^{\prime }+\left (a^{2} x \ln \left (x \right )^{2}+a \ln \left (x \right )+a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.137 |
|
|
\[
{}x y^{\prime \prime }+\left (f \left (x \right ) x +2\right ) y^{\prime }+f \left (x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.752 |
|
|
\[
{}\left (x -3\right ) y^{\prime \prime }-\left (4 x -9\right ) y^{\prime }+\left (3 x -6\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.111 |
|
|
\[
{}2 x y^{\prime \prime }+y^{\prime }+a y = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.639 |
|
|
\[
{}2 x y^{\prime \prime }-\left (-1+x \right ) y^{\prime }+a y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.740 |
|
|
\[
{}2 x y^{\prime \prime }-\left (2 x -1\right ) y^{\prime }+a y = 0
\] |
[_Laguerre] |
✗ |
0.745 |
|
|
\[
{}\left (2 x -1\right ) y^{\prime \prime }-\left (-4+3 x \right ) y^{\prime }+\left (x -3\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.897 |
|
|
\[
{}4 x y^{\prime \prime }-\left (x +a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.449 |
|
|
\[
{}4 x y^{\prime \prime }+2 y^{\prime }-y = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.063 |
|
|
\[
{}4 x y^{\prime \prime }+4 y^{\prime }-\left (x +2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.870 |
|
|
\[
{}4 x y^{\prime \prime }+4 y-\left (x +2\right ) y+l y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.491 |
|
|
\[
{}4 x y^{\prime \prime }+4 m y^{\prime }-\left (x -2 m -4 n \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.835 |
|
|
\[
{}16 x y^{\prime \prime }+8 y^{\prime }-\left (x +a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.694 |
|
|
\[
{}a x y^{\prime \prime }+b y^{\prime }+c y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.937 |
|
|
\[
{}a x y^{\prime \prime }+\left (b x +3 a \right ) y^{\prime }+3 b y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.224 |
|
|
\[
{}5 \left (a x +b \right ) y^{\prime \prime }+8 a y^{\prime }+c \left (a x +b \right )^{{1}/{5}} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.816 |
|
|
\[
{}2 a x y^{\prime \prime }+\left (b x +a \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.909 |
|
|
\[
{}2 a x y^{\prime \prime }+\left (b x +3 a \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.016 |
|
|
\[
{}\left (\operatorname {a2} x +\operatorname {b2} \right ) y^{\prime \prime }+\left (\operatorname {a1} x +\operatorname {b1} \right ) y^{\prime }+\left (\operatorname {a0} x +\operatorname {b0} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
7.620 |
|
|
\[
{}x^{2} y^{\prime \prime }-6 y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.447 |
|
|
\[
{}x^{2} y^{\prime \prime }-12 y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.443 |
|
|
\[
{}x^{2} y^{\prime \prime }+a y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.710 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a x +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.679 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (x^{2}-2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.572 |
|
|
\[
{}x^{2} y^{\prime \prime }-\left (a \,x^{2}+2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
4.204 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a^{2} x^{2}-6\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
3.541 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2}-v \left (v -1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.895 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.732 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{k}-b \left (b -1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.786 |
|
|
\[
{}x^{2} y^{\prime \prime }+\frac {y}{\ln \left (x \right )}-x \,{\mathrm e}^{x} \left (2+\ln \left (x \right ) x \right ) = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
0.332 |
|
|
\[
{}x^{2} y^{\prime \prime }+a y^{\prime }-x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.485 |
|
|
\[
{}x^{2} y^{\prime \prime }+a y^{\prime }-\left (b^{2} x^{2}+a b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.391 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }-y-a \,x^{2} = 0
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
1.704 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }+a y = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.402 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }-\left (x +a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.745 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }+\left (-v^{2}+x^{2}\right ) y = 0
\] |
[_Bessel] |
✓ |
0.871 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }+\left (-v^{2}+x^{2}\right ) y-f \left (x \right ) = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
0.903 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }+\left (l \,x^{2}-v^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.916 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (x +a \right ) y^{\prime }-y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.393 |
|
|
\[
{}x^{2} y^{\prime \prime }-x y^{\prime }+y-3 x^{3} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.619 |
|
|
\[
{}x^{2} y^{\prime \prime }-x y^{\prime }+\left (a \,x^{m}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.827 |
|
|
\[
{}x^{2} y^{\prime \prime }+2 x y^{\prime } = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.569 |
|
|
\[
{}x^{2} y^{\prime \prime }+2 x y^{\prime }+\left (a x -b^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.787 |
|
|
\[
{}x^{2} y^{\prime \prime }+2 x y^{\prime }+\left (a \,x^{2}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.858 |
|
|
\[
{}x^{2} y^{\prime \prime }+2 x y^{\prime }+\left (l \,x^{2}+a x -\left (n +1\right ) n \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.870 |
|
|
\[
{}x^{2} y^{\prime \prime }+2 \left (-1+x \right ) y^{\prime }+a y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.661 |
|
|
\[
{}x^{2} y^{\prime \prime }+2 \left (x +a \right ) y^{\prime }-b \left (b -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.770 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y-x^{5} \ln \left (x \right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.012 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x y^{\prime }-4 y-x \sin \left (x \right )-\left (a \,x^{2}+12 a +4\right ) \cos \left (x \right ) = 0
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
2.687 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.307 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y-\frac {x^{2}}{\cos \left (x \right )} = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.240 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y-\frac {x^{3}}{\cos \left (x \right )} = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
4.059 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (a^{2} x^{2}+2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.507 |
|
|
\[
{}x^{2} y^{\prime \prime }+3 x y^{\prime }+\left (-v^{2}+x^{2}+1\right ) y-f \left (x \right ) = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
0.907 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (3 x -1\right ) y^{\prime }+y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.301 |
|
|
\[
{}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y-5 x = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.364 |
|
|
\[
{}x^{2} y^{\prime \prime }-3 x y^{\prime }-5 y-x^{2} \ln \left (x \right ) = 0
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
1.566 |
|
|
\[
{}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y-x^{4}+x^{2} = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.876 |
|
|
\[
{}x^{2} y^{\prime \prime }+5 x y^{\prime }-\left (2 x^{3}-4\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.914 |
|
|
\[
{}x^{2} y^{\prime \prime }-5 x y^{\prime }+8 y-\sin \left (x \right ) x^{3} = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
2.349 |
|
|
\[
{}x^{2} y^{\prime \prime }+a x y^{\prime }+b y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
1.776 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.012 |
|
|
\[
{}x^{2} y^{\prime \prime }+a x y^{\prime }+\left (b \,x^{m}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.039 |
|
|
\[
{}x^{2} y^{\prime \prime }+x^{2} y^{\prime }+\left (a x +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.703 |
|
|
\[
{}x^{2} y^{\prime \prime }+x^{2} y^{\prime }-2 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.852 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (x^{2}-1\right ) y^{\prime }-y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.254 |
|
|
\[
{}x^{2} y^{\prime \prime }+x \left (x +1\right ) y^{\prime }+\left (x -9\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.969 |
|
|
\[
{}x^{2} y^{\prime \prime }+x \left (x +1\right ) y^{\prime }+\left (3 x -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.378 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (x +3\right ) x y^{\prime }-y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.677 |
|
|
\[
{}x^{2} y^{\prime \prime }-x \left (-1+x \right ) y^{\prime }+\left (-1+x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.355 |
|
|
\[
{}x^{2} y^{\prime \prime }-\left (x^{2}-2 x \right ) y^{\prime }-\left (x +a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.719 |
|
|
\[
{}x^{2} y^{\prime \prime }-\left (x^{2}-2 x \right ) y^{\prime }-\left (3 x +2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.109 |
|
|
\[
{}x^{2} y^{\prime \prime }-x \left (x +4\right ) y^{\prime }+4 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.018 |
|
|
\[
{}x^{2} y^{\prime \prime }+2 x^{2} y^{\prime }-v \left (v -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.742 |
|
|
\[
{}x^{2} y^{\prime \prime }+x \left (2 x +1\right ) y^{\prime }-4 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.961 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x \left (x +1\right ) y^{\prime }+2 \left (x +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.166 |
|
|
\[
{}x^{2} y^{\prime \prime }+a \,x^{2} y^{\prime }-2 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.990 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a +2 b \right ) x^{2} y^{\prime }+\left (\left (a +b \right ) b \,x^{2}-2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.220 |
|
|
\[
{}x^{2} y^{\prime \prime }+a \,x^{2} y^{\prime }+f \left (x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.488 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (2 a x +b \right ) x y^{\prime }+\left (a b x +c \,x^{2}+d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.644 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a x +b \right ) y^{\prime } x +\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x +\operatorname {c1} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.360 |
|
|
\[
{}x^{2} y^{\prime \prime }+x^{3} y^{\prime }+\left (x^{2}-2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.509 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (x^{2}+2\right ) x y^{\prime }+\left (x^{2}-2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.244 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x \left (x^{2}-a \right ) y^{\prime }+\left (2 n \,x^{2}+\left (\left (-1\right )^{n}-1\right ) a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.527 |
|
|
\[
{}x^{2} y^{\prime \prime }+4 x^{3} y^{\prime }+\left (4 x^{4}+2 x^{2}+1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.110 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2}+b \right ) x y^{\prime }+f \left (x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.890 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (x^{3}+1\right ) x y^{\prime }-y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.824 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (-x^{4}+\left (2 n +2 a +1\right ) x^{2}+\left (-1\right )^{n} a -a^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.114 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime } x +\left (\operatorname {a1} \,x^{2 n}+\operatorname {b1} \,x^{n}+\operatorname {c1} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.911 |
|
|
\[
{}x^{2} y^{\prime \prime }-\left (2 x^{2} \tan \left (x \right )-x \right ) y^{\prime }-\left (x \tan \left (x \right )+a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.464 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (2 x^{2} \cot \left (x \right )+x \right ) y^{\prime }+\left (x \cot \left (x \right )+a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.746 |
|
|
\[
{}x^{2} y^{\prime \prime }+2 x f \left (x \right ) y^{\prime }+\left (f^{\prime }\left (x \right ) x +f \left (x \right )^{2}-f \left (x \right )+a \,x^{2}+b x +c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.161 |
|