| # |
ODE |
CAS classification |
Solved? |
Maple |
Mma |
Sympy |
time(sec) |
| \begin{align*}
y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (a \,x^{3} b +a c \,x^{2}+b \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.207 |
|
| \begin{align*}
y^{\prime \prime }+\left (a \,x^{3}+2 b \right ) y^{\prime }+\left (a \,x^{3} b -a \,x^{2}+b^{2}\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.115 |
|
| \begin{align*}
y^{\prime \prime }+\left (a \,x^{3}+b x \right ) y^{\prime }+2 \left (2 a \,x^{2}+b \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.839 |
|
| \begin{align*}
y^{\prime \prime }+\left (a \,x^{3} b +b \,x^{2}+2 a \right ) y^{\prime }+a^{2} \left (b \,x^{3}+1\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.877 |
|
| \begin{align*}
y^{\prime \prime }+a \,x^{n} y^{\prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.591 |
|
| \begin{align*}
y^{\prime \prime }+a \,x^{n} y^{\prime }+b \,x^{n -1} y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
4.842 |
|
| \begin{align*}
y^{\prime \prime }+2 a \,x^{n} y^{\prime }+a \left (a \,x^{2 n}+n \,x^{n -1}\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.654 |
|
| \begin{align*}
y^{\prime \prime }+a \,x^{n} y^{\prime }+\left (b \,x^{2 n}+c \,x^{n -1}\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
4.374 |
|
| \begin{align*}
y^{\prime \prime }+a \,x^{n} y^{\prime }-b \left (a \,x^{n +m}+b \,x^{2 m}+m \,x^{m -1}\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
5.339 |
|
| \begin{align*}
y^{\prime \prime }+2 a \,x^{n} y^{\prime }+\left (a^{2} x^{2 n}+b \,x^{2 m}+x^{n -1} a n +c \,x^{m -1}\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
6.508 |
|
| \begin{align*}
y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+c \left (a \,x^{n}+b -c \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
5.601 |
|
| \begin{align*}
y^{\prime \prime }+\left (a \,x^{n}+2 b \right ) y^{\prime }+\left (x^{n} a b -a \,x^{n -1}+b^{2}\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
5.749 |
|
| \begin{align*}
y^{\prime \prime }+\left (x^{n} a b +b \,x^{n -1}+2 a \right ) y^{\prime }+a^{2} \left (b \,x^{n}+1\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
9.394 |
|
| \begin{align*}
y^{\prime \prime }+\left (x^{n} a b +2 b \,x^{n -1}-a^{2} x \right ) y^{\prime }+a \left (x^{n} a b +b \,x^{n -1}-a^{2} x \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
23.318 |
|
| \begin{align*}
y^{\prime \prime }+x^{n} \left (a \,x^{2}+\left (a c +b \right ) x +b c \right ) y^{\prime }-x^{n} \left (a x +b \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
25.927 |
|
| \begin{align*}
y^{\prime \prime }+\left (a \,x^{n}+x^{m} b \right ) y^{\prime }-\left (a \,x^{n -1}+x^{m -1} b \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.149 |
|
| \begin{align*}
y^{\prime \prime }+\left (a \,x^{n}+x^{m} b \right ) y^{\prime }+\left (x^{n -1} a n +b m \,x^{m -1}\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✗ |
1.312 |
|
| \begin{align*}
y^{\prime \prime }+\left (a \,x^{n}+x^{m} b \right ) y^{\prime }+\left (a \left (n +1\right ) x^{n -1}+b \left (m +1\right ) x^{m -1}\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
✗ |
8.474 |
|
| \begin{align*}
y^{\prime \prime }+\left (a \,x^{n}+x^{m} b \right ) y^{\prime }+c \left (a \,x^{n}+x^{m} b -c \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
6.987 |
|
| \begin{align*}
y^{\prime \prime }+\left (a \,x^{n}+x^{m} b \right ) y^{\prime }+\left (x^{n +m} a b +b \left (m +1\right ) x^{m -1}-a \,x^{n -1}\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
5.278 |
|
| \begin{align*}
y^{\prime \prime }+\left (a \,x^{n}+x^{m} b +c \right ) y^{\prime }+\left (x^{n +m} a b +x^{m} b c +x^{n -1} a n \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
7.246 |
|
| \begin{align*}
y^{\prime \prime } x +\frac {y^{\prime }}{2}+a y&=0 \\
\end{align*} |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✓ |
3.097 |
|
| \begin{align*}
b y+a y^{\prime }+y^{\prime \prime } x&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✗ |
0.913 |
|
| \begin{align*}
b x y+a y^{\prime }+y^{\prime \prime } x&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.545 |
|
| \begin{align*}
y^{\prime \prime } x +a y^{\prime }+\left (b x +c \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
6.082 |
|
| \begin{align*}
y^{\prime \prime } x +n y^{\prime }+b \,x^{1-2 n} y&=0 \\
\end{align*} |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✗ |
3.178 |
|
| \begin{align*}
y^{\prime \prime } x +\left (1-3 n \right ) y^{\prime }-a^{2} n^{2} x^{2 n -1} y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✗ |
✓ |
✓ |
✗ |
5.832 |
|
| \begin{align*}
y^{\prime \prime } x +a y^{\prime }+b \,x^{n} y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✗ |
0.793 |
|
| \begin{align*}
y^{\prime \prime } x +a y^{\prime }+b \,x^{n} \left (-b \,x^{n +1}+a +n \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
5.779 |
|
| \begin{align*}
y^{\prime \prime } x +a x y^{\prime }+a y&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✗ |
1.675 |
|
| \begin{align*}
y^{\prime \prime } x +\left (b -x \right ) y^{\prime }-a y&=0 \\
\end{align*} |
[_Laguerre] |
✗ |
✓ |
✓ |
✗ |
5.635 |
|
| \begin{align*}
y^{\prime \prime } x +\left (a x +b \right ) y^{\prime }+c \left (\left (a -c \right ) x +b \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
33.796 |
|
| \begin{align*}
y^{\prime \prime } x +\left (2 a x +b \right ) y^{\prime }+a \left (a x +b \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.964 |
|
| \begin{align*}
\left (a b x +a n +b m \right ) y+\left (m +n +\left (a +b \right ) x \right ) y^{\prime }+y^{\prime \prime } x&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
8.834 |
|
| \begin{align*}
y^{\prime \prime } x +\left (a x +b \right ) y^{\prime }+\left (c x +d \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
32.756 |
|
| \begin{align*}
y^{\prime \prime } x -\left (a x +1\right ) y^{\prime }-b \,x^{2} \left (b x +a \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.489 |
|
| \begin{align*}
y^{\prime \prime } x -\left (2 a x +1\right ) y^{\prime }+\left (b \,x^{3}+a^{2} x +a \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.955 |
|
| \begin{align*}
y^{\prime \prime } x +\left (a x +b \right ) y^{\prime }+c x \left (-c \,x^{2}+a x +b +1\right )&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✗ |
2.215 |
|
| \begin{align*}
y^{\prime \prime } x -\left (2 a x +1\right ) y^{\prime }+b \,x^{3} y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
6.542 |
|
| \begin{align*}
y^{\prime \prime } x +\left (a b \,x^{2}+b -5\right ) y^{\prime }+2 a^{2} \left (b -2\right ) x^{3} y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
38.014 |
|
| \begin{align*}
y^{\prime \prime } x +\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 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.559 |
|
| \begin{align*}
y^{\prime \prime } x +\left (a \,x^{2}+b x +2\right ) y^{\prime }+b y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.345 |
|
| \begin{align*}
y^{\prime \prime } x +\left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (2 a x +b \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✗ |
2.486 |
|
| \begin{align*}
y^{\prime \prime } x +\left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (c -1\right ) \left (a x +b \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.100 |
|
| \begin{align*}
y^{\prime \prime } x +\left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (A \,x^{2}+B x +\operatorname {C0} \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
24.050 |
|
| \begin{align*}
y^{\prime \prime } x +\left (a \,x^{2}+b x +2\right ) y^{\prime }+\left (c \,x^{2}+d x +b \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
23.521 |
|
| \begin{align*}
y^{\prime \prime } x +\left (a \,x^{3}+b \right ) y^{\prime }+a \left (b -1\right ) x^{2} y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
4.078 |
|
| \begin{align*}
y^{\prime \prime } x +\left (a \,x^{2}+b \right ) x y^{\prime }+\left (3 a \,x^{2}+b \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✗ |
1.951 |
|
| \begin{align*}
y^{\prime \prime } x +\left (a \,x^{3}+b \,x^{2}+2\right ) y^{\prime }+b x y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.839 |
|
| \begin{align*}
y^{\prime \prime } x +\left (a \,x^{3} b +b \,x^{2}+a x -1\right ) y^{\prime }+a^{2} b \,x^{3} y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.878 |
|
| \begin{align*}
y^{\prime \prime } x +\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 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.982 |
|
| \begin{align*}
y^{\prime \prime } x +a \,x^{n} y^{\prime }+\left (x^{n} a b -a \,x^{n -1}-b^{2} x +2 b \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
6.700 |
|
| \begin{align*}
y^{\prime \prime } x +\left (a \,x^{n}+2\right ) y^{\prime }+a \,x^{n -1} y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.379 |
|
| \begin{align*}
y^{\prime \prime } x +\left (x^{n}+1-n \right ) y^{\prime }+b \,x^{2 n -1} y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
8.874 |
|
| \begin{align*}
y^{\prime \prime } x +\left (a \,x^{n}+b \right ) y^{\prime }+x^{n -1} a n y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✗ |
1.533 |
|
| \begin{align*}
y^{\prime \prime } x +\left (a \,x^{n}+b \right ) y^{\prime }+a \left (b -1\right ) x^{n -1} y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.237 |
|
| \begin{align*}
y^{\prime \prime } x +\left (a \,x^{n}+b \right ) y^{\prime }+a \left (b +n -1\right ) x^{n -1} y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
3.998 |
|
| \begin{align*}
y^{\prime \prime } x +\left (a \,x^{n}+b \right ) y^{\prime }+c \left (a \,x^{n}-c x +b \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
9.065 |
|
| \begin{align*}
y^{\prime \prime } x +\left (x^{n} a b +b -3 n +1\right ) y^{\prime }+a^{2} n \left (b -n \right ) x^{2 n -1} y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
7.484 |
|
| \begin{align*}
y^{\prime \prime } x +\left (a \,x^{n}+b \right ) y^{\prime }+\left (c \,x^{2 n -1}+d \,x^{n -1}\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
5.301 |
|
| \begin{align*}
y^{\prime \prime } x +\left (a \,x^{n}+b \,x^{n -1}+2\right ) y^{\prime }+b \,x^{-2+n} y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
11.496 |
|
| \begin{align*}
y^{\prime \prime } x +\left (a \,x^{n}+b x \right ) y^{\prime }+\left (x^{n} a b +x^{n -1} a n -b \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
33.835 |
|
| \begin{align*}
y^{\prime \prime } x +\left (x^{n} a b +b \,x^{n -1}+a x -1\right ) y^{\prime }+a^{2} b \,x^{n} y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
29.232 |
|
| \begin{align*}
y^{\prime \prime } x +\left (a \,x^{n}+x^{m} b +c \right ) y^{\prime }+\left (c -1\right ) \left (a \,x^{n -1}+x^{m -1} b \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
✗ |
2.241 |
|
| \begin{align*}
y^{\prime \prime } x +\left (x^{n +m} a b +a n \,x^{n}+x^{m} b +1-2 n \right ) y^{\prime }+a^{2} b n \,x^{2 n +m -1} y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
11.078 |
|
| \begin{align*}
\left (a +x \right ) y^{\prime \prime }+\left (b x +c \right ) y^{\prime }+b y&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✗ |
4.007 |
|
| \begin{align*}
\left (a_{1} x +a_{0} \right ) y^{\prime \prime }+\left (b_{1} x +b_{0} \right ) y^{\prime }-m b_{1} y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
25.050 |
|
| \begin{align*}
\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 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
30.415 |
|
| \begin{align*}
\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 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
30.850 |
|
| \begin{align*}
\left (x +\gamma \right ) y^{\prime \prime }+\left (a \,x^{n}+x^{m} b +c \right ) y^{\prime }+\left (x^{n -1} a n +b m \,x^{m -1}\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✗ |
✗ |
3.230 |
|
| \begin{align*}
a y+x^{2} y^{\prime \prime }&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
0.680 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+\left (a x +b \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
0.526 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+\left (a^{2} x^{2}-n \left (n +1\right )\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
1.495 |
|
| \begin{align*}
-\left (n \left (n +1\right )+a^{2} x^{2}\right ) y+x^{2} y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
1.330 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-\left (a^{2} x^{2}+2 a b x +b^{2}-b \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.895 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
3.469 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-\left (a \,x^{3}+\frac {5}{16}\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
0.796 |
|
| \begin{align*}
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 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.174 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
0.495 |
|
| \begin{align*}
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 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
4.288 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+\left (a \,x^{2 n}+b \,x^{n}+c \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
2.795 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+\left (a \,x^{3 n}+b \,x^{2 n}+\frac {1}{4}-\frac {n^{2}}{4}\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
3.130 |
|
| \begin{align*}
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 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
3.965 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+a x y^{\prime }+b y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
3.452 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+y^{\prime } x +\left (x^{2}-\left (n +\frac {1}{2}\right )^{2}\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
5.463 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+y^{\prime } x -\left (x^{2}+\left (n +\frac {1}{2}\right )^{2}\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.668 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+y^{\prime } x +\left (-\nu ^{2}+x^{2}\right ) y&=0 \\
\end{align*} |
[_Bessel] |
✓ |
✓ |
✓ |
✓ |
1.845 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+y^{\prime } x -\left (\nu ^{2}+x^{2}\right ) y&=0 \\
\end{align*} |
[[_Bessel, _modified]] |
✓ |
✓ |
✓ |
✓ |
1.428 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+2 y^{\prime } x -\left (a^{2} x^{2}+2\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
1.843 |
|
| \begin{align*}
\left (a \left (1+a \right )+b^{2} x^{2}\right ) y-2 a x y^{\prime }+x^{2} y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.294 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 a x y^{\prime }+\left (-b^{2} x^{2}+a \left (1+a \right )\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.274 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+\lambda x y^{\prime }+\left (a \,x^{2}+b x +c \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
11.979 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+a x y^{\prime }+\left (b \,x^{n}+c \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.925 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+a x y^{\prime }+x^{n} \left (b \,x^{n}+c \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
9.997 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
21.666 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+a \,x^{2} y^{\prime }+\left (b \,x^{2}+c x +d \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
5.527 |
|
| \begin{align*}
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 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
25.119 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }-b y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
1.505 |
|
| \begin{align*}
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 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
27.791 |
|
| \begin{align*}
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 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
25.543 |
|