|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}4 x^{2} y^{\prime \prime }+4 y^{\prime } x +\left (-v^{2}+x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.741 |
|
|
\[
{}4 x^{2} y^{\prime \prime }+4 y^{\prime } x +\left (-x^{2}+2 \left (1-m +2 l \right ) x -m^{2}+1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.731 |
|
|
\[
{}4 x^{2} y^{\prime \prime }+4 y^{\prime } x -\left (4 x^{2}+1\right ) y-4 \sqrt {x^{3}}\, {\mathrm e}^{x} = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.639 |
|
|
\[
{}4 x^{2} y^{\prime \prime }+4 y^{\prime } x -\left (a \,x^{2}+1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.055 |
|
|
\[
{}4 x^{2} y^{\prime \prime }+4 y^{\prime } x +f \left (x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.224 |
|
|
\[
{}4 x^{2} y^{\prime \prime }+5 y^{\prime } x -y-\ln \left (x \right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.691 |
|
|
\[
{}4 x^{2} y^{\prime \prime }+8 y^{\prime } x -\left (4 x^{2}+12 x +3\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.102 |
|
|
\[
{}4 x^{2} y^{\prime \prime }-4 x \left (2 x -1\right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.885 |
|
|
\[
{}4 x^{2} y^{\prime \prime }+4 x^{3} y^{\prime }+\left (x^{2}+6\right ) \left (x^{2}-4\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.821 |
|
|
\[
{}4 x^{2} y^{\prime \prime }+4 x^{2} \ln \left (x \right ) y^{\prime }+\left (x^{2} \ln \left (x \right )^{2}+2 x -8\right ) y-4 x^{2} \sqrt {{\mathrm e}^{x} x^{-x}} = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.445 |
|
|
\[
{}\left (2 x +1\right )^{2} y^{\prime \prime }-2 \left (2 x +1\right ) y^{\prime }-12 y-3 x -1 = 0
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
1.165 |
|
|
\[
{}x \left (4 x -1\right ) y^{\prime \prime }+\left (\left (4 a +2\right ) x -a \right ) y^{\prime }+a \left (a -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.017 |
|
|
\[
{}\left (3 x -1\right )^{2} y^{\prime \prime }+3 \left (3 x -1\right ) y^{\prime }-9 y-\ln \left (3 x -1\right )^{2} = 0
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
2.265 |
|
|
\[
{}9 x \left (x -1\right ) y^{\prime \prime }+3 \left (2 x -1\right ) y^{\prime }-20 y = 0
\] |
[_Jacobi] |
✓ |
1.318 |
|
|
\[
{}16 x^{2} y^{\prime \prime }+\left (4 x +3\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.890 |
|
|
\[
{}16 x^{2} y^{\prime \prime }+32 y^{\prime } x -\left (4 x +5\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
3.735 |
|
|
\[
{}\left (27 x^{2}+4\right ) y^{\prime \prime }+27 y^{\prime } x -3 y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.982 |
|
|
\[
{}48 x \left (x -1\right ) y^{\prime \prime }+\left (152 x -40\right ) y^{\prime }+53 y = 0
\] |
[_Jacobi] |
✗ |
1.063 |
|
|
\[
{}50 x \left (x -1\right ) y^{\prime \prime }+25 \left (2 x -1\right ) y^{\prime }-2 y = 0
\] |
[_Jacobi, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
2.705 |
|
|
\[
{}144 x \left (x -1\right ) y^{\prime \prime }+\left (120 x -48\right ) y^{\prime }+y = 0
\] |
[_Jacobi] |
✗ |
0.954 |
|
|
\[
{}144 x \left (x -1\right ) y^{\prime \prime }+\left (168 x -96\right ) y^{\prime }+y = 0
\] |
[_Jacobi] |
✗ |
0.731 |
|
|
\[
{}a \,x^{2} y^{\prime \prime }+b x y^{\prime }+\left (c \,x^{2}+d x +f \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.804 |
|
|
\[
{}\operatorname {a2} \,x^{2} y^{\prime \prime }+\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x \right ) y^{\prime }+\left (\operatorname {a0} \,x^{2}+\operatorname {b0} x +\operatorname {c0} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.209 |
|
|
\[
{}\left (a \,x^{2}+1\right ) y^{\prime \prime }+a x y^{\prime }+b y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
281.598 |
|
|
\[
{}\left (a^{2} x^{2}-1\right ) y^{\prime \prime }+2 a^{2} x y^{\prime } = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.211 |
|
|
\[
{}\left (a^{2} x^{2}-1\right ) y^{\prime \prime }+2 a^{2} x y^{\prime }-2 a^{2} y = 0
\] |
[_Gegenbauer] |
✓ |
1.664 |
|
|
\[
{}\left (a \,x^{2}+b x \right ) y^{\prime \prime }+2 b y^{\prime }-2 a y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.381 |
|
|
\[
{}\operatorname {A2} \left (a x +b \right )^{2} y^{\prime \prime }+\operatorname {A1} \left (a x +b \right ) y^{\prime }+\operatorname {A0} \left (a x +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.546 |
|
|
\[
{}\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (d x +f \right ) y^{\prime }+g y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
3.448 |
|
|
\[
{}x^{3} y^{\prime \prime }+y^{\prime } x -\left (2 x +3\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.347 |
|
|
\[
{}x^{3} y^{\prime \prime }+2 y^{\prime } x -y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.346 |
|
|
\[
{}x^{3} y^{\prime \prime }+x^{2} y^{\prime }+\left (a \,x^{2}+b x +a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.305 |
|
|
\[
{}x^{3} y^{\prime \prime }+x \left (x +1\right ) y^{\prime }-2 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.277 |
|
|
\[
{}x^{3} y^{\prime \prime }-x^{2} y^{\prime }+y x -\ln \left (x \right )^{3} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.461 |
|
|
\[
{}x^{3} y^{\prime \prime }-\left (x^{2}-1\right ) y^{\prime }+y x = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.547 |
|
|
\[
{}x^{3} y^{\prime \prime }+3 x^{2} y^{\prime }+y x -1 = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.348 |
|
|
\[
{}x \left (x^{2}+1\right ) y^{\prime \prime }+\left (2 x^{2}+1\right ) y^{\prime }-v \left (v +1\right ) x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.889 |
|
|
\[
{}x \left (x^{2}+1\right ) y^{\prime \prime }+2 \left (x^{2}-1\right ) y^{\prime }-2 y x = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.116 |
|
|
\[
{}x \left (x^{2}+1\right ) y^{\prime \prime }+\left (2 \left (n +1\right ) x^{2}+2 n +1\right ) y^{\prime }-\left (v -n \right ) \left (v +n +1\right ) x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.124 |
|
|
\[
{}x \left (x^{2}+1\right ) y^{\prime \prime }-\left (2 \left (n -1\right ) x^{2}+2 n -1\right ) y^{\prime }+\left (v +n \right ) \left (-v +n -1\right ) x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.115 |
|
|
\[
{}x \left (x^{2}-1\right ) y^{\prime \prime }+y^{\prime }+y a \,x^{3} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
2.350 |
|
|
\[
{}x \left (x^{2}-1\right ) y^{\prime \prime }+\left (x^{2}-1\right ) y^{\prime }-y x = 0
\] |
[[_elliptic, _class_II]] |
✗ |
106.811 |
|
|
\[
{}x \left (x^{2}-1\right ) y^{\prime \prime }+\left (3 x^{2}-1\right ) y^{\prime }+y x = 0
\] |
[[_elliptic, _class_I]] |
✗ |
0.683 |
|
|
\[
{}x \left (x^{2}-1\right ) y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.308 |
|
|
\[
{}x \left (x^{2}+2\right ) y^{\prime \prime }-y^{\prime }-6 y x = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.782 |
|
|
\[
{}x \left (x^{2}-2\right ) y^{\prime \prime }-\left (x^{3}+3 x^{2}-2 x -2\right ) y^{\prime }+\left (x^{2}+4 x +2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.490 |
|
|
\[
{}x^{2} \left (x +1\right ) y^{\prime \prime }-x \left (2 x +1\right ) y^{\prime }+\left (2 x +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.075 |
|
|
\[
{}x^{2} \left (x +1\right ) y^{\prime \prime }+2 x \left (2+3 x \right ) y^{\prime } = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.985 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 \left (x -2\right ) y^{\prime }}{x \left (x -1\right )}+\frac {2 \left (x +1\right ) y}{x^{2} \left (x -1\right )}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.033 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (5 x -4\right ) y^{\prime }}{x \left (x -1\right )}-\frac {\left (9 x -6\right ) y}{x^{2} \left (x -1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.172 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (\left (a +b +1\right ) x +\alpha +\beta -1\right ) y^{\prime }}{x \left (x -1\right )}-\frac {\left (a b x -\alpha \beta \right ) y}{x^{2} \left (x -1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.413 |
|
|
\[
{}y^{\prime \prime } = -\frac {y^{\prime }}{x +1}-\frac {y}{x \left (x +1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.871 |
|
|
\[
{}y^{\prime \prime } = \frac {2 y^{\prime }}{x \left (x -2\right )}-\frac {y}{x^{2} \left (x -2\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
83.686 |
|
|
\[
{}y^{\prime \prime } = \frac {2 y}{x \left (x -1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.893 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (\left (\alpha +\beta +1\right ) x^{2}-\left (\alpha +\beta +1+a \left (\gamma +\delta \right )-\delta \right ) x +a \gamma \right ) y^{\prime }}{x \left (x -1\right ) \left (x -a \right )}-\frac {\left (\alpha \beta x -q \right ) y}{x \left (x -1\right ) \left (x -a \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.129 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (A \,x^{2}+B x +C \right ) y^{\prime }}{\left (x -a \right ) \left (x -b \right ) \left (x -c \right )}-\frac {\left (\operatorname {DD} x +E \right ) y}{\left (x -a \right ) \left (x -b \right ) \left (x -c \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.697 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (x -4\right ) y^{\prime }}{2 x \left (x -2\right )}-\frac {\left (x -3\right ) y}{2 x^{2} \left (x -2\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.121 |
|
|
\[
{}y^{\prime \prime } = \frac {y^{\prime }}{x +1}-\frac {\left (3 x +1\right ) y}{4 x^{2} \left (x +1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.073 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (x -1\right )}+\frac {v \left (v +1\right ) y}{4 x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.765 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (\left (a +1\right ) x -1\right ) y^{\prime }}{x \left (x -1\right )}-\frac {\left (\left (a^{2}-b^{2}\right ) x +c^{2}\right ) y}{4 x^{2} \left (x -1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.090 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (x -1\right )}-\frac {\left (a x +b \right ) y}{4 x \left (x -1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.715 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (-3 x +1\right ) y}{\left (x -1\right ) \left (2 x -1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.029 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (3 x +a +2 b \right ) y^{\prime }}{2 \left (x +a \right ) \left (x +b \right )}-\frac {\left (a -b \right ) y}{4 \left (x +a \right )^{2} \left (x +b \right )}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
2.373 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (6 x -1\right ) y^{\prime }}{3 x \left (x -2\right )}+\frac {y}{3 x^{2} \left (x -2\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.299 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (a \left (b +2\right ) x^{2}+\left (c -d +1\right ) x \right ) y^{\prime }}{\left (a x +1\right ) x^{2}}-\frac {\left (a b x -c d \right ) y}{\left (a x +1\right ) x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.326 |
|
|
\[
{}y^{\prime \prime } = \frac {2 \left (a x +2 b \right ) y^{\prime }}{x \left (a x +b \right )}-\frac {\left (2 a x +6 b \right ) y}{\left (a x +b \right ) x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.025 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (2 a x +b \right ) y^{\prime }}{x \left (a x +b \right )}-\frac {\left (a v x -b \right ) y}{\left (a x +b \right ) x^{2}}+A x
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
107.698 |
|
|
\[
{}y^{\prime \prime } = -\frac {a y}{x^{4}}
\] |
[[_Emden, _Fowler]] |
✓ |
3.264 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (x^{2} a \left (1-a \right )-b \left (x +b \right )\right ) y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.480 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left ({\mathrm e}^{\frac {2}{x}}-v^{2}\right ) y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.169 |
|
|
\[
{}y^{\prime \prime } = -\frac {y^{\prime }}{x^{3}}+\frac {2 y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.265 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (a +b \right ) y^{\prime }}{x^{2}}-\frac {\left (\left (a +b \right ) x +a b \right ) y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.456 |
|
|
\[
{}y^{\prime \prime } = -\frac {y^{\prime }}{x}-\frac {y}{x^{4}}
\] |
[[_Emden, _Fowler]] |
✓ |
0.761 |
|
|
\[
{}y^{\prime \prime } = -\frac {y^{\prime }}{x}-\frac {\left (b \,x^{2}+a \left (x^{4}+1\right )\right ) y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.683 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.557 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 y^{\prime }}{x}-\frac {a^{2} y}{x^{4}}
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.125 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x^{3}}+\frac {y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.300 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 \left (x +a \right ) y^{\prime }}{x^{2}}-\frac {b y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.573 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (2 x^{2}-1\right ) y^{\prime }}{x^{3}}-\frac {y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.210 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (2 x^{2}-1\right ) y^{\prime }}{x^{3}}-\frac {2 y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.657 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (x^{3}-1\right ) y^{\prime }}{x \left (x^{3}+1\right )}+\frac {x y}{x^{3}+1}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.075 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x \left (x^{2}+1\right )}-\frac {\left (-v \left (v +1\right ) x^{2}-n^{2}\right ) y}{x^{2} \left (x^{2}+1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.977 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (a \,x^{2}+a -1\right ) y^{\prime }}{x \left (x^{2}+1\right )}-\frac {\left (b \,x^{2}+c \right ) y}{x^{2} \left (x^{2}+1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.086 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (x^{2}-2\right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {\left (x^{2}-2\right ) y}{x^{2} \left (x^{2}-1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.220 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {v \left (v +1\right ) y}{x^{2} \left (x^{2}-1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.835 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}+\frac {v \left (v +1\right ) y}{x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.846 |
|
|
\[
{}y^{\prime \prime } = \frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (a \left (a +1\right )-a \,x^{2} \left (a +3\right )\right ) y}{x^{2} \left (x^{2}-1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.641 |
|
|
\[
{}x^{2} \left (x^{2}-1\right ) y^{\prime \prime }-2 x^{3} y^{\prime }-\left (\left (a -n \right ) \left (a +n +1\right ) x^{2} \left (x^{2}-1\right )+2 a \,x^{2}+n \left (n +1\right ) \left (x^{2}-1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.961 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (a \,x^{2}+a -2\right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {b y}{x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.375 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (2 b c \,x^{c} \left (x^{2}-1\right )+2 \left (a -1\right ) x^{2}-2 a \right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {\left (b^{2} c^{2} x^{2 c} \left (x^{2}-1\right )+b c \,x^{c +2} \left (2 a -c -1\right )-b c \,x^{c} \left (2 a -c +1\right )+x^{2} \left (a \left (a -1\right )-v \left (v +1\right )\right )-a \left (a +1\right )\right ) y}{x^{2} \left (x^{2}-1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.747 |
|
|
\[
{}y^{\prime \prime } = -\frac {a y}{\left (x^{2}+1\right )^{2}}
\] |
[_Halm] |
✓ |
1.359 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}+1}-\frac {y}{\left (x^{2}+1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.046 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}+1}-\frac {\left (a^{2} \left (x^{2}+1\right )^{2}-n \left (n +1\right ) \left (x^{2}+1\right )+m^{2}\right ) y}{\left (x^{2}+1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.991 |
|
|
\[
{}y^{\prime \prime } = -\frac {a x y^{\prime }}{x^{2}+1}-\frac {b y}{\left (x^{2}+1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.927 |
|
|
\[
{}y^{\prime \prime } = -\frac {a y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.461 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}+\frac {a^{2} y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.404 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (-a^{2}-\lambda \left (x^{2}-1\right )\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.804 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (\left (x^{2}-1\right ) \left (a \,x^{2}+b x +c \right )-k^{2}\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.932 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (-a^{2} \left (x^{2}-1\right )^{2}-n \left (n +1\right ) \left (x^{2}-1\right )-m^{2}\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.991 |
|
|
\[
{}y^{\prime \prime } = \frac {2 x \left (2 a -1\right ) y^{\prime }}{x^{2}-1}-\frac {\left (x^{2} \left (2 a \left (2 a -1\right )-v \left (v +1\right )\right )+2 a +v \left (v +1\right )\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.099 |
|