|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}x y^{\prime \prime }+\left (a \,x^{n}+b x \right ) y^{\prime }+\left (a b \,x^{n}+a n \,x^{n -1}-b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.796 |
|
|
\[
{}x y^{\prime \prime }+\left (a b \,x^{n}+b \,x^{n -1}+a x -1\right ) y^{\prime }+a^{2} b \,x^{n} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.278 |
|
|
\[
{}x y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime }+\left (c -1\right ) \left (a \,x^{n -1}+b \,x^{m -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.602 |
|
|
\[
{}x y^{\prime \prime }+\left (a b \,x^{n +m}+a n \,x^{n}+b \,x^{m}+1-2 n \right ) y^{\prime }+a^{2} b n \,x^{2 n +m -1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.389 |
|
|
\[
{}\left (x +a \right ) y^{\prime \prime }+\left (b x +c \right ) y^{\prime }+b y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.642 |
|
|
\[
{}\left (a_{1} x +a_{0} \right ) y^{\prime \prime }+\left (b_{1} x +b_{0} \right ) y^{\prime }-m b_{1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.263 |
|
|
\[
{}\left (a x +b \right ) y^{\prime \prime }+s \left (c x +d \right ) y^{\prime }-s^{2} \left (\left (a +c \right ) x +b +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.758 |
|
|
\[
{}\left (a_{2} x +b_{2} \right ) y^{\prime \prime }+\left (a_{1} x +b_{1} \right ) y^{\prime }+\left (a_{0} x +b_{0} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
6.104 |
|
|
\[
{}\left (x +\gamma \right ) y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime }+\left (a n \,x^{n -1}+b m \,x^{m -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.568 |
|
|
\[
{}x^{2} y^{\prime \prime }+a y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
1.101 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a x +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.832 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a^{2} x^{2}-n \left (n +1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.053 |
|
|
\[
{}x^{2} y^{\prime \prime }-\left (a^{2} x^{2}+n \left (n +1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.152 |
|
|
\[
{}x^{2} y^{\prime \prime }-\left (a^{2} x^{2}+2 a b x +b^{2}-b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.633 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.634 |
|
|
\[
{}x^{2} y^{\prime \prime }-\left (a \,x^{3}+\frac {5}{16}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
4.631 |
|
|
\[
{}x^{2} y^{\prime \prime }-\left (a^{2} x^{4}+a \left (2 b -1\right ) x^{2}+b \left (b +1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.684 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.990 |
|
|
\[
{}x^{2} y^{\prime \prime }-\left (a^{2} x^{2 n}+a \left (2 b +n -1\right ) x^{n}+b \left (b -1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.333 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2 n}+b \,x^{n}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.294 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{3 n}+b \,x^{2 n}+\frac {1}{4}-\frac {n^{2}}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.322 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2 n} \left (b \,x^{n}+c \right )^{m}+\frac {1}{4}-\frac {n^{2}}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.583 |
|
|
\[
{}x^{2} y^{\prime \prime }+a x y^{\prime }+b y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
1.863 |
|
|
\[
{}x^{2} y^{\prime \prime }+y^{\prime } x +\left (x^{2}-\left (n +\frac {1}{2}\right )^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.106 |
|
|
\[
{}x^{2} y^{\prime \prime }+y^{\prime } x -\left (x^{2}+\left (n +\frac {1}{2}\right )^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.131 |
|
|
\[
{}x^{2} y^{\prime \prime }+y^{\prime } x +\left (-\nu ^{2}+x^{2}\right ) y = 0
\] |
[_Bessel] |
✓ |
1.148 |
|
|
\[
{}x^{2} y^{\prime \prime }+y^{\prime } x -\left (\nu ^{2}+x^{2}\right ) y = 0
\] |
[[_Bessel, _modified]] |
✓ |
1.148 |
|
|
\[
{}x^{2} y^{\prime \prime }+2 y^{\prime } x -\left (a^{2} x^{2}+2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
4.204 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 a x y^{\prime }+\left (b^{2} x^{2}+a \left (a +1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.618 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 a x y^{\prime }+\left (-b^{2} x^{2}+a \left (a +1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.544 |
|
|
\[
{}x^{2} y^{\prime \prime }+\lambda x y^{\prime }+\left (a \,x^{2}+b x +c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.936 |
|
|
\[
{}x^{2} y^{\prime \prime }+a x y^{\prime }+\left (b \,x^{n}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.238 |
|
|
\[
{}x^{2} y^{\prime \prime }+a x y^{\prime }+x^{n} \left (b \,x^{n}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.503 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.898 |
|
|
\[
{}x^{2} y^{\prime \prime }+a \,x^{2} y^{\prime }+\left (b \,x^{2}+c x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.057 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{2}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.403 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }-b y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.381 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (k \left (a -k \right ) x^{2}+\left (a n +b k -2 k n \right ) x +n \left (b -n -1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.491 |
|
|
\[
{}a_{2} x^{2} y^{\prime \prime }+\left (a_{1} x^{2}+b_{1} x \right ) y^{\prime }+\left (a_{0} x^{2}+b_{0} x +c_{0} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.311 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2}+\left (a b -1\right ) x +b \right ) y^{\prime }+a^{2} b x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
11.389 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x \left (x^{2}-a \right ) y^{\prime }+\left (2 n \,x^{2}+\left (\left (-1\right )^{n}-1\right ) a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.397 |
|
|
\[
{}x^{2} y^{\prime \prime }+x \left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (A \,x^{3}+B \,x^{2}+C x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.150 |
|
|
\[
{}x^{2} y^{\prime \prime }+a \,x^{n} y^{\prime }-\left (a b \,x^{n}+a c \,x^{n -1}+b^{2} x^{2}+2 b c x +c^{2}-c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.193 |
|
|
\[
{}x^{2} y^{\prime \prime }+a \,x^{n} y^{\prime }+\left (a b \,x^{n +2 m}-b^{2} x^{4 m +2}+a m \,x^{n -1}-m^{2}-m \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.354 |
|
|
\[
{}x^{2} y^{\prime \prime }+x \left (a \,x^{n}+b \right ) y^{\prime }+b \left (a \,x^{n}-1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.953 |
|
|
\[
{}x^{2} y^{\prime \prime }+x \left (a \,x^{n}+b \right ) y^{\prime }+\left (\alpha \,x^{2 n}+\beta \,x^{n}+\gamma \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.848 |
|
|
\[
{}x^{2} y^{\prime \prime }+x \left (2 a \,x^{n}+b \right ) y^{\prime }+\left (a^{2} x^{2 n}+a \left (b +n -1\right ) x^{n}+\alpha \,x^{2 m}+\beta \,x^{m}+\gamma \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.071 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{n +2}+b \,x^{2}+c \right ) y^{\prime }+\left (a n \,x^{n +1}+a c \,x^{n}+b c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.039 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+n \left (n -1\right ) y = 0
\] |
[_Gegenbauer] |
✗ |
0.532 |
|
|
\[
{}\left (-a^{2}+x^{2}\right ) y^{\prime \prime }+b y^{\prime }-6 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.216 |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }+y^{\prime } x +a y = 0
\] |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
2.950 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x +n^{2} y = 0
\] |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.699 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +n \left (n +1\right ) y = 0
\] |
[_Gegenbauer] |
✗ |
0.895 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +\nu \left (\nu +1\right ) y = 0
\] |
[_Gegenbauer] |
✗ |
0.904 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-3 y^{\prime } x +n \left (n +2\right ) y = 0
\] |
[_Gegenbauer] |
✓ |
1.956 |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }+2 \left (n +1\right ) x y^{\prime }-\left (\nu +n +1\right ) \left (\nu -n \right ) y = 0
\] |
[_Gegenbauer] |
✗ |
1.250 |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }-2 \left (n -1\right ) x y^{\prime }-\left (\nu -n +1\right ) \left (\nu +n \right ) y = 0
\] |
[_Gegenbauer] |
✗ |
1.257 |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }+\left (2 a +1\right ) y^{\prime }-b \left (2 a +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.063 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (2 a -3\right ) x y^{\prime }+\left (n +1\right ) \left (n +2 a -1\right ) y = 0
\] |
[_Gegenbauer] |
✗ |
1.237 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (\beta -\alpha -\left (\alpha +\beta +2\right ) x \right ) y^{\prime }+n \left (n +\alpha +\beta +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.403 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (\alpha -\beta +\left (\alpha +\beta -2\right ) x \right ) y^{\prime }+\left (n +1\right ) \left (n +\alpha +\beta \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.395 |
|
|
\[
{}\left (a \,x^{2}+b \right ) y^{\prime \prime }+a x y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
313.301 |
|
|
\[
{}\left (x^{2}+a \right ) y^{\prime \prime }+2 b x y^{\prime }+2 \left (b -1\right ) y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.760 |
|
|
\[
{}\left (-a^{2}+x^{2}\right ) y^{\prime \prime }+2 b x y^{\prime }+b \left (b -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
63.261 |
|
|
\[
{}\left (a^{2}+x^{2}\right ) y^{\prime \prime }+2 b x y^{\prime }+b \left (b -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.051 |
|
|
\[
{}\left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (2 n +1\right ) a x y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
424.660 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x +\left (2 a \,x^{2}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.962 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.197 |
|
|
\[
{}\left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (c \,x^{2}+d \right ) y^{\prime }+\lambda \left (\left (-a \lambda +c \right ) x^{2}+d -b \lambda \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.113 |
|
|
\[
{}\left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (\lambda \left (a +c \right ) x^{2}+\left (c -a \right ) x +2 b \lambda \right ) y^{\prime }+\lambda ^{2} \left (c \,x^{2}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.675 |
|
|
\[
{}x \left (x -1\right ) y^{\prime \prime }+\left (\left (\alpha +\beta +1\right ) x -\gamma \right ) y^{\prime }+\alpha \beta y = 0
\] |
[_Jacobi] |
✗ |
1.418 |
|
|
\[
{}x \left (x +a \right ) y^{\prime \prime }+\left (b x +c \right ) y^{\prime }+d y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.287 |
|
|
\[
{}2 x \left (x -1\right ) y^{\prime \prime }+\left (2 x -1\right ) y^{\prime }+\left (a x +b \right ) y = 0
\] |
[_Jacobi] |
✗ |
0.659 |
|
|
\[
{}\left (2 a x +x^{2}+b \right ) y^{\prime \prime }+\left (x +a \right ) y^{\prime }-m^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.865 |
|
|
\[
{}\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (d x +k \right ) y^{\prime }+\left (d -2 a \right ) y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
35.191 |
|
|
\[
{}\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (k x +d \right ) y^{\prime }-k y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
3.898 |
|
|
\[
{}\left (a \,x^{2}+2 b x +c \right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+d y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
6.810 |
|
|
\[
{}\left (a \,x^{2}+2 b x +c \right ) y^{\prime \prime }+3 \left (a x +b \right ) y^{\prime }+d y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
5.689 |
|
|
\[
{}\left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime \prime }+\left (b_{1} x +c_{1} \right ) y^{\prime }+c_{0} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
3.637 |
|
|
\[
{}\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }-\left (-k^{2}+x^{2}\right ) y^{\prime }+\left (x +k \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
3.220 |
|
|
\[
{}\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (k^{3}+x^{3}\right ) y^{\prime }-\left (k^{2}-k x +x^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
4.172 |
|
|
\[
{}x^{3} y^{\prime \prime }+\left (a x +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.813 |
|
|
\[
{}x^{3} y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.080 |
|
|
\[
{}x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+b y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.901 |
|
|
\[
{}x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.946 |
|
|
\[
{}x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (c x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.029 |
|
|
\[
{}x^{3} y^{\prime \prime }+\left (a \,x^{3}+a b x -x^{2}+b \right ) y^{\prime }+a^{2} b x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.079 |
|
|
\[
{}x^{3} y^{\prime \prime }+x \left (a \,x^{n}+b \right ) y^{\prime }-\left (a \,x^{n}-a b \,x^{n -1}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.885 |
|
|
\[
{}x \left (a \,x^{2}+b \right ) y^{\prime \prime }+2 \left (a \,x^{2}+b \right ) y^{\prime }-2 y a x = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
4.546 |
|
|
\[
{}x \left (x^{2}+a \right ) y^{\prime \prime }+\left (b \,x^{2}+c \right ) y^{\prime }+s x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.573 |
|
|
\[
{}x^{2} \left (a x +b \right ) y^{\prime \prime }+\left (c \,x^{2}+\left (a \lambda +2 b \right ) x +b \lambda \right ) y^{\prime }+\lambda \left (c -2 a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.543 |
|
|
\[
{}x^{2} \left (a x +b \right ) y^{\prime \prime }-2 x \left (a x +2 b \right ) y^{\prime }+2 \left (a x +3 b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.042 |
|
|
\[
{}x^{2} \left (a x +b \right ) y^{\prime \prime }+\left (a \left (2-n -m \right ) x^{2}-b \left (n +m \right ) x \right ) y^{\prime }+\left (a m \left (n -1\right ) x +b n \left (m +1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.619 |
|
|
\[
{}x^{2} \left (x +a_{2} \right ) y^{\prime \prime }+x \left (b_{1} x +a_{1} \right ) y^{\prime }+\left (b_{0} x +a_{0} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.402 |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +2 c \right ) y^{\prime }+\left (\beta -2 b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
4.372 |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +2 c \right ) y^{\prime }-\left (\alpha x +2 b -\beta \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
4.401 |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (-2 a \,x^{2}-\left (b +1\right ) x +k \right ) y^{\prime }+2 \left (a x +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
3.781 |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (n \,x^{2}+m x +k \right ) y^{\prime }+\left (k -1\right ) \left (\left (-a k +n \right ) x +m -b k \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
4.694 |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\left (m -a \right ) x^{2}+\left (2 c m -1\right ) x -c \right ) y^{\prime }+\left (-2 m x +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
504.900 |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (n \,x^{2}+m x +k \right ) y^{\prime }+\left (-2 \left (a +n \right ) x +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
52.479 |
|