|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}x y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }-\left (a c \,x^{2}+\left (b c +c^{2}+a \right ) x +b +2 c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.932 |
|
|
\[
{}x y^{\prime \prime }+\left (a \,x^{2}+b x +2\right ) y^{\prime }+b y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.835 |
|
|
\[
{}x y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (2 a x +b \right ) y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.116 |
|
|
\[
{}x y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (c -1\right ) \left (a x +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.655 |
|
|
\[
{}x y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (A \,x^{2}+B x +\operatorname {C0} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.548 |
|
|
\[
{}x y^{\prime \prime }+\left (a \,x^{2}+b x +2\right ) y^{\prime }+\left (c \,x^{2}+d x +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.413 |
|
|
\[
{}x y^{\prime \prime }+\left (a \,x^{3}+b \right ) y^{\prime }+a \left (b -1\right ) x^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.797 |
|
|
\[
{}x y^{\prime \prime }+\left (a \,x^{2}+b \right ) x y^{\prime }+\left (3 a \,x^{2}+b \right ) y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.178 |
|
|
\[
{}x y^{\prime \prime }+\left (a \,x^{3}+b \,x^{2}+2\right ) y^{\prime }+b x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.940 |
|
|
\[
{}x y^{\prime \prime }+\left (a \,x^{3} b +b \,x^{2}+a x -1\right ) y^{\prime }+a^{2} b \,x^{3} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.891 |
|
|
\[
{}x y^{\prime \prime }+\left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime }+\left (d -1\right ) \left (a \,x^{2}+b x +c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.005 |
|
|
\[
{}x y^{\prime \prime }+a \,x^{n} y^{\prime }+\left (a b \,x^{n}-a \,x^{n -1}-b^{2} x +2 b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.283 |
|
|
\[
{}x y^{\prime \prime }+\left (a \,x^{n}+2\right ) y^{\prime }+a \,x^{n -1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.339 |
|
|
\[
{}x y^{\prime \prime }+\left (x^{n}+1-n \right ) y^{\prime }+b \,x^{2 n -1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.115 |
|
|
\[
{}x y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+a n \,x^{n -1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.121 |
|
|
\[
{}x y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+a \left (b -1\right ) x^{n -1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.398 |
|
|
\[
{}x y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+a \left (b +n -1\right ) x^{n -1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.491 |
|
|
\[
{}x y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+c \left (a \,x^{n}-c x +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.365 |
|
|
\[
{}x y^{\prime \prime }+\left (a b \,x^{n}+b -3 n +1\right ) y^{\prime }+a^{2} n \left (b -n \right ) x^{2 n -1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.327 |
|
|
\[
{}x y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+\left (c \,x^{2 n -1}+d \,x^{n -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.265 |
|
|
\[
{}x y^{\prime \prime }+\left (a \,x^{n}+b \,x^{n -1}+2\right ) y^{\prime }+b \,x^{n -2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.494 |
|
|
\[
{}x y^{\prime \prime }+\left (a \,x^{n}+b x \right ) y^{\prime }+\left (a b \,x^{n}+a n \,x^{n -1}-b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.289 |
|
|
\[
{}x y^{\prime \prime }+\left (a b \,x^{n}+b \,x^{n -1}+a x -1\right ) y^{\prime }+a^{2} b \,x^{n} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.880 |
|
|
\[
{}x y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime }+\left (c -1\right ) \left (a \,x^{n -1}+b \,x^{m -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.447 |
|
|
\[
{}x y^{\prime \prime }+\left (a b \,x^{n +m}+a n \,x^{n}+b \,x^{m}+1-2 n \right ) y^{\prime }+a^{2} b n \,x^{2 n +m -1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.086 |
|
|
\[
{}\left (x +a \right ) y^{\prime \prime }+\left (b x +c \right ) y^{\prime }+b y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.232 |
|
|
\[
{}\left (a_{1} x +a_{0} \right ) y^{\prime \prime }+\left (b_{1} x +b_{0} \right ) y^{\prime }-m b_{1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.812 |
|
|
\[
{}\left (a x +b \right ) y^{\prime \prime }+s \left (c x +d \right ) y^{\prime }-s^{2} \left (\left (a +c \right ) x +b +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.934 |
|
|
\[
{}\left (a_{2} x +b_{2} \right ) y^{\prime \prime }+\left (a_{1} x +b_{1} \right ) y^{\prime }+\left (a_{0} x +b_{0} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
8.679 |
|
|
\[
{}\left (x +\gamma \right ) y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime }+\left (a n \,x^{n -1}+b m \,x^{m -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
0.986 |
|
|
\[
{}x^{2} y^{\prime \prime }+a y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.591 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a x +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.352 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a^{2} x^{2}-n \left (n +1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.656 |
|
|
\[
{}x^{2} y^{\prime \prime }-\left (a^{2} x^{2}+n \left (n +1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.654 |
|
|
\[
{}x^{2} y^{\prime \prime }-\left (a^{2} x^{2}+2 a b x +b^{2}-b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.001 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.984 |
|
|
\[
{}x^{2} y^{\prime \prime }-\left (a \,x^{3}+\frac {5}{16}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.600 |
|
|
\[
{}x^{2} y^{\prime \prime }-\left (a^{2} x^{4}+a \left (2 b -1\right ) x^{2}+b \left (b +1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.128 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.331 |
|
|
\[
{}x^{2} y^{\prime \prime }-\left (a^{2} x^{2 n}+a \left (2 b +n -1\right ) x^{n}+b \left (b -1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.654 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2 n}+b \,x^{n}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.700 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{3 n}+b \,x^{2 n}+\frac {1}{4}-\frac {n^{2}}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.587 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2 n} \left (b \,x^{n}+c \right )^{m}+\frac {1}{4}-\frac {n^{2}}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.917 |
|
|
\[
{}x^{2} y^{\prime \prime }+a x y^{\prime }+b y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
1.778 |
|
|
\[
{}x^{2} y^{\prime \prime }+y^{\prime } x +\left (x^{2}-\left (n +\frac {1}{2}\right )^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.838 |
|
|
\[
{}x^{2} y^{\prime \prime }+y^{\prime } x -\left (x^{2}+\left (n +\frac {1}{2}\right )^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.714 |
|
|
\[
{}x^{2} y^{\prime \prime }+y^{\prime } x +\left (-\nu ^{2}+x^{2}\right ) y = 0
\] |
[_Bessel] |
✓ |
0.671 |
|
|
\[
{}x^{2} y^{\prime \prime }+y^{\prime } x -\left (\nu ^{2}+x^{2}\right ) y = 0
\] |
[[_Bessel, _modified]] |
✓ |
0.700 |
|
|
\[
{}x^{2} y^{\prime \prime }+2 y^{\prime } x -\left (a^{2} x^{2}+2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.735 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 a x y^{\prime }+\left (b^{2} x^{2}+a \left (a +1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.798 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 a x y^{\prime }+\left (-b^{2} x^{2}+a \left (a +1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.767 |
|
|
\[
{}x^{2} y^{\prime \prime }+\lambda x y^{\prime }+\left (a \,x^{2}+b x +c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.476 |
|
|
\[
{}x^{2} y^{\prime \prime }+a x y^{\prime }+\left (b \,x^{n}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.485 |
|
|
\[
{}x^{2} y^{\prime \prime }+a x y^{\prime }+x^{n} \left (b \,x^{n}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.884 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.646 |
|
|
\[
{}x^{2} y^{\prime \prime }+a \,x^{2} y^{\prime }+\left (b \,x^{2}+c x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.478 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{2}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.978 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }-b y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.820 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (k \left (a -k \right ) x^{2}+\left (a n +b k -2 k n \right ) x +n \left (b -n -1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.277 |
|
|
\[
{}a_{2} x^{2} y^{\prime \prime }+\left (a_{1} x^{2}+b_{1} x \right ) y^{\prime }+\left (a_{0} x^{2}+b_{0} x +c_{0} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
3.453 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2}+\left (a b -1\right ) x +b \right ) y^{\prime }+a^{2} b x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.476 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x \left (x^{2}-a \right ) y^{\prime }+\left (2 n \,x^{2}+\left (\left (-1\right )^{n}-1\right ) a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.988 |
|
|
\[
{}x^{2} y^{\prime \prime }+x \left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (A \,x^{3}+B \,x^{2}+C x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.852 |
|
|
\[
{}x^{2} y^{\prime \prime }+a \,x^{n} y^{\prime }-\left (a b \,x^{n}+a c \,x^{n -1}+b^{2} x^{2}+2 b x c +c^{2}-c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.740 |
|
|
\[
{}x^{2} y^{\prime \prime }+a \,x^{n} y^{\prime }+\left (a b \,x^{n +2 m}-b^{2} x^{4 m +2}+a m \,x^{n -1}-m^{2}-m \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.905 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime } x +b \left (a \,x^{n}-1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.634 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime } x +\left (\alpha \,x^{2 n}+\beta \,x^{n}+\gamma \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.314 |
|
|
\[
{}x^{2} y^{\prime \prime }+x \left (2 a \,x^{n}+b \right ) y^{\prime }+\left (a^{2} x^{2 n}+a \left (b +n -1\right ) x^{n}+\alpha \,x^{2 m}+\beta \,x^{m}+\gamma \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.495 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{n +2}+b \,x^{2}+c \right ) y^{\prime }+\left (a n \,x^{n +1}+a c \,x^{n}+b c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.622 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+n \left (n -1\right ) y = 0
\] |
[_Gegenbauer] |
✗ |
0.962 |
|
|
\[
{}\left (-a^{2}+x^{2}\right ) y^{\prime \prime }+b y^{\prime }-6 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.769 |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }+y^{\prime } x +a y = 0
\] |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.985 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x +n^{2} y = 0
\] |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.058 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +n \left (n +1\right ) y = 0
\] |
[_Gegenbauer] |
✗ |
1.351 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +\nu \left (\nu +1\right ) y = 0
\] |
[_Gegenbauer] |
✗ |
1.385 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-3 y^{\prime } x +n \left (n +2\right ) y = 0
\] |
[_Gegenbauer] |
✓ |
0.749 |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }+2 \left (n +1\right ) x y^{\prime }-\left (\nu +n +1\right ) \left (\nu -n \right ) y = 0
\] |
[_Gegenbauer] |
✗ |
1.787 |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }-2 \left (n -1\right ) x y^{\prime }-\left (\nu -n +1\right ) \left (\nu +n \right ) y = 0
\] |
[_Gegenbauer] |
✗ |
1.908 |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }+\left (2 a +1\right ) y^{\prime }-b \left (2 a +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.718 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (2 a -3\right ) x y^{\prime }+\left (n +1\right ) \left (n +2 a -1\right ) y = 0
\] |
[_Gegenbauer] |
✗ |
1.954 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (\beta -\alpha -\left (\alpha +\beta +2\right ) x \right ) y^{\prime }+n \left (n +\alpha +\beta +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.205 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (\alpha -\beta +\left (\alpha +\beta -2\right ) x \right ) y^{\prime }+\left (n +1\right ) \left (n +\alpha +\beta \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.191 |
|
|
\[
{}\left (a \,x^{2}+b \right ) y^{\prime \prime }+a x y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
172.314 |
|
|
\[
{}\left (x^{2}+a \right ) y^{\prime \prime }+2 b x y^{\prime }+2 \left (b -1\right ) y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.500 |
|
|
\[
{}\left (-a^{2}+x^{2}\right ) y^{\prime \prime }+2 b x y^{\prime }+b \left (b -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
31.610 |
|
|
\[
{}\left (a^{2}+x^{2}\right ) y^{\prime \prime }+2 b x y^{\prime }+b \left (b -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.095 |
|
|
\[
{}\left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (2 n +1\right ) a x y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
524.549 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x +\left (2 a \,x^{2}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.605 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.904 |
|
|
\[
{}\left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (c \,x^{2}+d \right ) y^{\prime }+\lambda \left (\left (-a \lambda +c \right ) x^{2}+d -b \lambda \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.835 |
|
|
\[
{}\left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (\lambda \left (a +c \right ) x^{2}+\left (c -a \right ) x +2 b \lambda \right ) y^{\prime }+\lambda ^{2} \left (c \,x^{2}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
4.036 |
|
|
\[
{}x \left (x -1\right ) y^{\prime \prime }+\left (\left (\alpha +\beta +1\right ) x -\gamma \right ) y^{\prime }+\alpha \beta y = 0
\] |
[_Jacobi] |
✗ |
2.370 |
|
|
\[
{}x \left (x +a \right ) y^{\prime \prime }+\left (b x +c \right ) y^{\prime }+d y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.010 |
|
|
\[
{}2 x \left (x -1\right ) y^{\prime \prime }+\left (2 x -1\right ) y^{\prime }+\left (a x +b \right ) y = 0
\] |
[_Jacobi] |
✗ |
1.194 |
|
|
\[
{}\left (2 a x +x^{2}+b \right ) y^{\prime \prime }+\left (x +a \right ) y^{\prime }-m^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.466 |
|
|
\[
{}\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (d x +k \right ) y^{\prime }+\left (d -2 a \right ) y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
19.480 |
|
|
\[
{}\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (k x +d \right ) y^{\prime }-k y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
4.238 |
|
|
\[
{}\left (a \,x^{2}+2 b x +c \right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+d y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
3.259 |
|
|
\[
{}\left (a \,x^{2}+2 b x +c \right ) y^{\prime \prime }+3 \left (a x +b \right ) y^{\prime }+d y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.511 |
|
|
\[
{}\left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime \prime }+\left (b_{1} x +c_{1} \right ) y^{\prime }+c_{0} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
3.894 |
|