|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }-\left (x^{2}+\left (n +\frac {1}{2}\right )^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.942 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }+\left (-\nu ^{2}+x^{2}\right ) y = 0
\] |
[_Bessel] |
✓ |
2.134 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }-\left (\nu ^{2}+x^{2}\right ) y = 0
\] |
[[_Bessel, _modified]] |
✓ |
2.237 |
|
|
\[
{}x^{2} y^{\prime \prime }+2 x y^{\prime }-\left (a^{2} x^{2}+2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
5.665 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 a x y^{\prime }+\left (b^{2} x^{2}+a \left (a +1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
3.118 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 a x y^{\prime }+\left (-b^{2} x^{2}+a \left (a +1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.939 |
|
|
\[
{}x^{2} y^{\prime \prime }+\lambda x y^{\prime }+\left (x^{2} a +b x +c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.962 |
|
|
\[
{}x^{2} y^{\prime \prime }+a x y^{\prime }+\left (b \,x^{n}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
4.463 |
|
|
\[
{}x^{2} y^{\prime \prime }+a x y^{\prime }+x^{n} \left (b \,x^{n}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.516 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.968 |
|
|
\[
{}x^{2} y^{\prime \prime }+a \,x^{2} y^{\prime }+\left (b \,x^{2}+c x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.076 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (x^{2} a +b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{2}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.442 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (x^{2} a +b x \right ) y^{\prime }-b y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.316 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (x^{2} a +b x \right ) y^{\prime }+\left (k \left (a -k \right ) x^{2}+\left (a n +b k -2 k n \right ) x +n \left (-n +b -1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.519 |
|
|
\[
{}a_{2} x^{2} y^{\prime \prime }+\left (a_{1} x^{2}+b_{1} x \right ) y^{\prime }+\left (a_{0} x^{2}+b_{0} x +c_{0} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.386 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (x^{2} a +\left (a b -1\right ) x +b \right ) y^{\prime }+a^{2} b x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
12.573 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x \left (x^{2}-a \right ) y^{\prime }+\left (2 n \,x^{2}+\left (\left (-1\right )^{n}-1\right ) a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.447 |
|
|
\[
{}x^{2} y^{\prime \prime }+x \left (x^{2} a +b x +c \right ) y^{\prime }+\left (A \,x^{3}+B \,x^{2}+C x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.186 |
|
|
\[
{}x^{2} y^{\prime \prime }+a \,x^{n} y^{\prime }-\left (a b \,x^{n}+a c \,x^{n -1}+b^{2} x^{2}+2 b c x +c^{2}-c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.169 |
|
|
\[
{}x^{2} y^{\prime \prime }+a \,x^{n} y^{\prime }+\left (a b \,x^{n +2 m}-b^{2} x^{4 m +2}+a m \,x^{n -1}-m^{2}-m \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.191 |
|
|
\[
{}x^{2} y^{\prime \prime }+x \left (a \,x^{n}+b \right ) y^{\prime }+b \left (a \,x^{n}-1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.918 |
|
|
\[
{}x^{2} y^{\prime \prime }+x \left (a \,x^{n}+b \right ) y^{\prime }+\left (\alpha \,x^{2 n}+\beta \,x^{n}+\gamma \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.887 |
|
|
\[
{}x^{2} y^{\prime \prime }+x \left (2 a \,x^{n}+b \right ) y^{\prime }+\left (a^{2} x^{2 n}+a \left (n +b -1\right ) x^{n}+\alpha \,x^{2 m}+\beta \,x^{m}+\gamma \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.113 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2+n}+b \,x^{2}+c \right ) y^{\prime }+\left (a n \,x^{n +1}+a c \,x^{n}+b c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.952 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+n \left (n -1\right ) y = 0
\] |
[_Gegenbauer] |
✗ |
0.554 |
|
|
\[
{}\left (-a^{2}+x^{2}\right ) y^{\prime \prime }+b y^{\prime }-6 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.260 |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }+x y^{\prime }+a y = 0
\] |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
2.447 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+n^{2} y = 0
\] |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.736 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+n \left (n +1\right ) y = 0
\] |
[_Gegenbauer] |
✗ |
0.935 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+\nu \left (\nu +1\right ) y = 0
\] |
[_Gegenbauer] |
✗ |
0.934 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-3 x y^{\prime }+n \left (2+n \right ) y = 0
\] |
[_Gegenbauer] |
✓ |
2.073 |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }+2 \left (n +1\right ) x y^{\prime }-\left (\nu +n +1\right ) \left (\nu -n \right ) y = 0
\] |
[_Gegenbauer] |
✗ |
1.284 |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }-2 \left (n -1\right ) x y^{\prime }-\left (\nu -n +1\right ) \left (\nu +n \right ) y = 0
\] |
[_Gegenbauer] |
✗ |
1.299 |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }+\left (2 a +1\right ) y^{\prime }-b \left (2 a +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.098 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (2 a -3\right ) x y^{\prime }+\left (n +1\right ) \left (n +2 a -1\right ) y = 0
\] |
[_Gegenbauer] |
✗ |
1.286 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (\beta -\alpha -\left (\alpha +\beta +2\right ) x \right ) y^{\prime }+n \left (n +\alpha +\beta +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.461 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (\alpha -\beta +\left (\alpha +\beta -2\right ) x \right ) y^{\prime }+\left (n +1\right ) \left (n +\alpha +\beta \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.433 |
|
|
\[
{}\left (x^{2} a +b \right ) y^{\prime \prime }+a x y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
271.231 |
|
|
\[
{}\left (x^{2}+a \right ) y^{\prime \prime }+2 b x y^{\prime }+2 \left (b -1\right ) y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.799 |
|
|
\[
{}\left (-a^{2}+x^{2}\right ) y^{\prime \prime }+2 b x y^{\prime }+b \left (b -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
64.161 |
|
|
\[
{}\left (a^{2}+x^{2}\right ) y^{\prime \prime }+2 b x y^{\prime }+b \left (b -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.201 |
|
|
\[
{}\left (x^{2} a +b \right ) y^{\prime \prime }+\left (2 n +1\right ) a x y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
420.820 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+\left (2 x^{2} a +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.000 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.275 |
|
|
\[
{}\left (x^{2} a +b \right ) y^{\prime \prime }+\left (c \,x^{2}+d \right ) y^{\prime }+\lambda \left (\left (-a \lambda +c \right ) x^{2}+d -b \lambda \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.178 |
|
|
\[
{}\left (x^{2} a +b \right ) y^{\prime \prime }+\left (\lambda \left (a +c \right ) x^{2}+\left (c -a \right ) x +2 b \lambda \right ) y^{\prime }+\lambda ^{2} \left (c \,x^{2}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.688 |
|
|
\[
{}x \left (x -1\right ) y^{\prime \prime }+\left (\left (\alpha +\beta +1\right ) x -\gamma \right ) y^{\prime }+\alpha \beta y = 0
\] |
[_Jacobi] |
✗ |
1.466 |
|
|
\[
{}x \left (x +a \right ) y^{\prime \prime }+\left (b x +c \right ) y^{\prime }+d y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.332 |
|
|
\[
{}2 x \left (x -1\right ) y^{\prime \prime }+\left (2 x -1\right ) y^{\prime }+\left (a x +b \right ) y = 0
\] |
[_Jacobi] |
✗ |
0.678 |
|
|
\[
{}\left (2 a x +x^{2}+b \right ) y^{\prime \prime }+\left (x +a \right ) y^{\prime }-m^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.939 |
|
|
\[
{}\left (x^{2} a +b x +c \right ) y^{\prime \prime }+\left (d x +k \right ) y^{\prime }+\left (d -2 a \right ) y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
36.026 |
|
|
\[
{}\left (x^{2} a +b x +c \right ) y^{\prime \prime }+\left (k x +d \right ) y^{\prime }-k y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
4.059 |
|
|
\[
{}\left (x^{2} a +2 b x +c \right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+d y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
5.984 |
|
|
\[
{}\left (x^{2} a +2 b x +c \right ) y^{\prime \prime }+3 \left (a x +b \right ) y^{\prime }+d y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
5.890 |
|
|
\[
{}\left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime \prime }+\left (b_{1} x +c_{1} \right ) y^{\prime }+c_{0} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
3.743 |
|
|
\[
{}\left (x^{2} a +b x +c \right ) y^{\prime \prime }-\left (-k^{2}+x^{2}\right ) y^{\prime }+\left (x +k \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
3.378 |
|
|
\[
{}\left (x^{2} a +b x +c \right ) y^{\prime \prime }+\left (k^{3}+x^{3}\right ) y^{\prime }-\left (k^{2}-k x +x^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
4.224 |
|
|
\[
{}x^{3} y^{\prime \prime }+\left (a x +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.841 |
|
|
\[
{}x^{3} y^{\prime \prime }+\left (x^{2} a +b \right ) y^{\prime }+c x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.101 |
|
|
\[
{}x^{3} y^{\prime \prime }+\left (x^{2} a +b x \right ) y^{\prime }+b y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.923 |
|
|
\[
{}x^{3} y^{\prime \prime }+\left (x^{2} a +b x \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.964 |
|
|
\[
{}x^{3} y^{\prime \prime }+\left (x^{2} a +b x \right ) y^{\prime }+\left (c x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.036 |
|
|
\[
{}x^{3} y^{\prime \prime }+\left (a \,x^{3}+a b x -x^{2}+b \right ) y^{\prime }+a^{2} b x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.162 |
|
|
\[
{}x^{3} y^{\prime \prime }+x \left (a \,x^{n}+b \right ) y^{\prime }-\left (a \,x^{n}-a b \,x^{n -1}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.784 |
|
|
\[
{}x \left (x^{2} a +b \right ) y^{\prime \prime }+2 \left (x^{2} a +b \right ) y^{\prime }-2 a x y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
4.045 |
|
|
\[
{}x \left (x^{2}+a \right ) y^{\prime \prime }+\left (b \,x^{2}+c \right ) y^{\prime }+s x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.631 |
|
|
\[
{}x^{2} \left (a x +b \right ) y^{\prime \prime }+\left (c \,x^{2}+\left (a \lambda +2 b \right ) x +b \lambda \right ) y^{\prime }+\lambda \left (c -2 a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.585 |
|
|
\[
{}x^{2} \left (a x +b \right ) y^{\prime \prime }-2 x \left (a x +2 b \right ) y^{\prime }+2 \left (a x +3 b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.080 |
|
|
\[
{}x^{2} \left (a x +b \right ) y^{\prime \prime }+\left (a \left (2-n -m \right ) x^{2}-b \left (m +n \right ) x \right ) y^{\prime }+\left (a m \left (n -1\right ) x +b n \left (m +1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.663 |
|
|
\[
{}x^{2} \left (x +a_{2} \right ) y^{\prime \prime }+x \left (b_{1} x +a_{1} \right ) y^{\prime }+\left (b_{0} x +a_{0} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.454 |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +2 c \right ) y^{\prime }+\left (\beta -2 b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
4.533 |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +2 c \right ) y^{\prime }-\left (\alpha x +2 b -\beta \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
4.538 |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (-2 x^{2} a -\left (b +1\right ) x +k \right ) y^{\prime }+2 \left (a x +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
3.839 |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (n \,x^{2}+m x +k \right ) y^{\prime }+\left (k -1\right ) \left (\left (-a k +n \right ) x +m -b k \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
4.524 |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\left (m -a \right ) x^{2}+\left (2 c m -1\right ) x -c \right ) y^{\prime }+\left (-2 m x +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
532.500 |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (n \,x^{2}+m x +k \right ) y^{\prime }+\left (-2 \left (a +n \right ) x +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
52.614 |
|
|
\[
{}\left (a \,x^{3}+x^{2}+b \right ) y^{\prime \prime }+a^{2} x \left (x^{2}-b \right ) y^{\prime }-a^{3} b x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
9.914 |
|
|
\[
{}2 x \left (x^{2} a +b x +c \right ) y^{\prime \prime }+\left (x^{2} a -c \right ) y^{\prime }+\lambda \,x^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
112.509 |
|
|
\[
{}x \left (x^{2} a +b x +1\right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y^{\prime }+\left (n x +m \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
4.222 |
|
|
\[
{}x \left (x -1\right ) \left (x -a \right ) y^{\prime \prime }+\left (\left (\alpha +\beta +1\right ) x^{2}-\left (\alpha +\beta +1+a \left (\gamma +d \right )-a \right ) x +a \gamma \right ) y^{\prime }+\left (\alpha \beta x -q \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.622 |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }-\left (-\lambda ^{2}+x^{2}\right ) y^{\prime }+\left (x +\lambda \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
164.282 |
|
|
\[
{}2 \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+\left (3 x^{2} a +2 b x +c \right ) y^{\prime }+\lambda y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
164.718 |
|
|
\[
{}2 \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+3 \left (3 x^{2} a +2 b x +c \right ) y^{\prime }+\left (6 a x +2 b +\lambda \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
272.195 |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\left (\alpha \gamma +\beta \right ) x +\beta \lambda \right ) y^{\prime }-\left (\alpha x +\beta \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
483.006 |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+\left (\lambda ^{3}+x^{3}\right ) y^{\prime }-\left (\lambda ^{2}-\lambda x +x^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
475.401 |
|
|
\[
{}2 x \left (x^{2} a +b x +c \right ) y^{\prime \prime }+\left (a \left (2-k \right ) x^{2}+b \left (1-k \right ) x -c k \right ) y^{\prime }+\lambda \,x^{k +1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.138 |
|
|
\[
{}x^{4} y^{\prime \prime }+a y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
3.467 |
|
|
\[
{}x^{4} y^{\prime \prime }+\left (x^{2} a +b x +c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.600 |
|
|
\[
{}x^{4} y^{\prime \prime }-\left (a +b \right ) x^{2} y^{\prime }+\left (x \left (a +b \right )+a b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.902 |
|
|
\[
{}x^{4} y^{\prime \prime }+2 x^{2} \left (x +a \right ) y^{\prime }+b y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.073 |
|
|
\[
{}x^{4} y^{\prime \prime }+a \,x^{n} y^{\prime }-\left (a \,x^{n -1}+a b \,x^{n -2}+b^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.200 |
|
|
\[
{}x^{2} \left (x -a \right )^{2} y^{\prime \prime }+b y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.072 |
|
|
\[
{}x^{2} \left (x -a \right )^{2} y^{\prime \prime }+b y = c \,x^{2} \left (x -a \right )^{2}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
2.298 |
|
|
\[
{}a \,x^{2} \left (x -1\right )^{2} y^{\prime \prime }+\left (b \,x^{2}+c x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.849 |
|
|
\[
{}x^{2} \left (x^{2}+a \right ) y^{\prime \prime }+\left (b \,x^{2}+c \right ) x y^{\prime }+d y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.666 |
|
|
\[
{}\left (x^{2}+1\right )^{2} y^{\prime \prime }+a y = 0
\] |
[_Halm] |
✓ |
1.487 |
|
|
\[
{}\left (x^{2}-1\right )^{2} y^{\prime \prime }+a y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.635 |
|
|
\[
{}\left (a^{2}+x^{2}\right )^{2} y^{\prime \prime }+b^{2} y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
1.830 |
|
|
\[
{}\left (-a^{2}+x^{2}\right )^{2} y^{\prime \prime }+b^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.068 |
|
|
\[
{}4 \left (x^{2}+1\right )^{2} y^{\prime \prime }+\left (x^{2} a +a -3\right ) y = 0
\] |
[_Halm] |
✓ |
1.625 |
|