| # |
ODE |
CAS classification |
Solved? |
Maple |
Mma |
Sympy |
time(sec) |
| \begin{align*}
y^{\prime } y+\frac {a \left (x -6\right ) y}{5 x^{{7}/{5}}}&=\frac {2 a^{2} \left (x -1\right ) \left (x +4\right )}{5 x^{{9}/{5}}} \\
\end{align*} |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✓ |
✗ |
✗ |
63.458 |
|
| \begin{align*}
y^{\prime } y-\frac {3 a y}{x^{{7}/{4}}}&=\frac {a^{2} \left (x -1\right ) \left (x -9\right )}{4 x^{{5}/{2}}} \\
\end{align*} |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
✗ |
✗ |
78.745 |
|
| \begin{align*}
y^{\prime } y-\frac {a \left (\left (1+k \right ) x -1\right ) y}{x^{2}}&=\frac {a^{2} \left (1+k \right ) \left (x -1\right )}{x^{2}} \\
\end{align*} |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✓ |
✗ |
✗ |
42.982 |
|
| \begin{align*}
y^{\prime } y-\left (\left (2 n -1\right ) x -a n \right ) x^{-1-n} y&=n \left (x -a \right ) x^{-2 n} \\
\end{align*} |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✓ |
✗ |
✗ |
113.552 |
|
| \begin{align*}
y^{\prime } y-a \left (\frac {n +2}{n}+b \,x^{n}\right ) y&=-\frac {a^{2} x \left (\frac {n +1}{n}+b \,x^{n}\right )}{n} \\
\end{align*} |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✓ |
✗ |
✗ |
143.492 |
|
| \begin{align*}
y^{\prime } y&=\left (a \,{\mathrm e}^{x}+b \right ) y+c \,{\mathrm e}^{2 x}-a b \,{\mathrm e}^{x}-b^{2} \\
\end{align*} |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✓ |
✗ |
✗ |
42.887 |
|
| \begin{align*}
y^{\prime } y&=\left (a \,{\mathrm e}^{\lambda x}+b \right ) y+c \left (a^{2} {\mathrm e}^{2 \lambda x}+a b \left (\lambda x +1\right ) {\mathrm e}^{\lambda x}+b^{2} \lambda x \right ) \\
\end{align*} |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✓ |
✓ |
✗ |
124.284 |
|
| \begin{align*}
y^{\prime } y&={\mathrm e}^{\lambda x} \left (2 a \lambda x +a +b \right ) y-{\mathrm e}^{2 \lambda x} \left (a^{2} \lambda \,x^{2}+a b x +c \right ) \\
\end{align*} |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✓ |
✗ |
✗ |
193.800 |
|
| \begin{align*}
y^{\prime } y&={\mathrm e}^{a x} \left (2 a \,x^{2}+b +2 x \right ) y+{\mathrm e}^{2 a x} \left (-a \,x^{4}-b \,x^{2}+c \right ) \\
\end{align*} |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
✗ |
✗ |
95.198 |
|
| \begin{align*}
y^{\prime } y+a \left (2 b x +1\right ) {\mathrm e}^{b x} y&=-a^{2} b \,x^{2} {\mathrm e}^{2 b x} \\
\end{align*} |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✓ |
✓ |
✗ |
14.368 |
|
| \begin{align*}
y^{\prime } y-a \left (1+2 n +2 n \left (n +1\right ) x \right ) {\mathrm e}^{\left (n +1\right ) x} y&=-a^{2} n \left (n +1\right ) \left (n x +1\right ) x \,{\mathrm e}^{2 \left (n +1\right ) x} \\
\end{align*} |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✓ |
✗ |
✗ |
81.632 |
|
| \begin{align*}
y^{\prime } y+a \left (1+2 b \sqrt {x}\right ) {\mathrm e}^{2 b \sqrt {x}} y&=-a^{2} b \,x^{{3}/{2}} {\mathrm e}^{4 b \sqrt {x}} \\
\end{align*} |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✓ |
✗ |
✗ |
33.967 |
|
| \begin{align*}
y^{\prime } y&=\left (2 \ln \left (x \right )+a +1\right ) y+x \left (-\ln \left (x \right )^{2}-a \ln \left (x \right )+b \right ) \\
\end{align*} |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✓ |
✗ |
✗ |
47.377 |
|
| \begin{align*}
y^{\prime } y&=\left (2 \ln \left (x \right )^{2}+2 \ln \left (x \right )+a \right ) y+x \left (-\ln \left (x \right )^{4}-a \ln \left (x \right )^{2}+b \right ) \\
\end{align*} |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
✗ |
✗ |
68.372 |
|
| \begin{align*}
y^{\prime } y&=a x \cos \left (\lambda \,x^{2}\right ) y+x \\
\end{align*} |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
✗ |
✗ |
32.331 |
|
| \begin{align*}
\left (A y+B x +a \right ) y^{\prime }+B y+k x +b&=0 \\
\end{align*} |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
✓ |
✗ |
9.609 |
|
| \begin{align*}
\left (y+a x +b \right ) y^{\prime }&=\alpha y+\beta x +\gamma \\
\end{align*} | [[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] | ✓ | ✓ | ✗ | ✗ | 52.945 |
|
| \begin{align*}
\left (y+A \,x^{n}+a \right ) y^{\prime }+n A \,x^{n -1} y+k \,x^{m}+b&=0 \\
\end{align*} |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
✓ |
✗ |
33.041 |
|
| \begin{align*}
x y^{\prime } y&=a y^{2}+b y+c \,x^{n}+s \\
\end{align*} |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
✗ |
✗ |
66.837 |
|
| \begin{align*}
x y^{\prime } y&=-n y^{2}+a \left (2 n +1\right ) x y+b y-a^{2} n \,x^{2}-a b x +c \\
\end{align*} |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✓ |
✗ |
✗ |
37.094 |
|
| \begin{align*}
2 x y^{\prime } y&=\left (1-n \right ) y^{2}+\left (a \left (2 n +1\right ) x +2 n -1\right ) y-a^{2} n \,x^{2}-b x -n \\
\end{align*} |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✓ |
✗ |
✗ |
54.204 |
|
| \begin{align*}
\left (a x y-a k y+b x -b k \right ) y^{\prime }&=c y^{2}+d x y+\left (-d k +b \right ) y \\
\end{align*} |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✓ |
✗ |
✗ |
69.732 |
|
| \begin{align*}
x \left (2 a y+b x \right ) y^{\prime }&=a \left (2-m \right ) y^{2}+b \left (1-m \right ) x y+c \,x^{2}+A \,x^{m +2} \\
\end{align*} |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
✓ |
✗ |
6.036 |
|
| \begin{align*}
\left (x^{2}+y x +a \right ) y^{\prime }&=y^{2}+y x +b \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
✓ |
✓ |
8.730 |
|
| \begin{align*}
\left (2 A x y+B \,x^{2}+b \right ) y^{\prime }&=A y^{2}+k \left (A k +B \right ) x^{2}+c \\
\end{align*} |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
✓ |
✗ |
88.516 |
|
| \begin{align*}
\left (A x y+B \,x^{2}+k x \right ) y^{\prime }&=A y^{2}+B y x +\left (A b +k \right ) y+B b x +b k \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
0.133 |
|
| \begin{align*}
\left (A x y+B \,x^{2}+k x \right ) y^{\prime }&=A y^{2}+c x y+d \,x^{2}+\left (-A \beta +k \right ) y-c \beta x -k \beta \\
\end{align*} |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✓ |
✗ |
✗ |
103.514 |
|
| \begin{align*}
\left (A x y+A k y+B \,x^{2}+B k x \right ) y^{\prime }&=c y^{2}+d x y+k \left (d -B \right ) y \\
\end{align*} |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✓ |
✗ |
✗ |
121.221 |
|
| \begin{align*}
\left (A x y+B \,x^{2}+\left (-1+k \right ) A a y-\left (A b k +B a \right ) x \right ) y^{\prime }&=A y^{2}+B y x -\left (B a k +A b \right ) y+\left (-1+k \right ) B b x \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
✓ |
✗ |
7.464 |
|
| \begin{align*}
\left (\left (a x +c \right ) y+\left (1-n \right ) x^{2}+\left (2 n -1\right ) x -n \right ) y^{\prime }&=2 a y^{2}+2 y x \\
\end{align*} |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✓ |
✗ |
✗ |
272.934 |
|
| \begin{align*}
x \left (2 a x y+b \right ) y^{\prime }&=-a \left (m +3\right ) x y^{2}-b \left (m +2\right ) y+c \,x^{m} \\
\end{align*} |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
✓ |
✗ |
5.849 |
|
| \begin{align*}
x \left (\left (m -1\right ) \left (A x +B \right ) y+m \left (d \,x^{2}+e x +F \right )\right ) y^{\prime }&=\left (A \left (1-n \right ) x -B n \right ) y^{2}+\left (d \left (2-n \right ) x^{2}+e \left (1-n \right ) x -F n \right ) y \\
\end{align*} |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✓ |
✓ |
✗ |
426.141 |
|
| \begin{align*}
x \left (2 a x y+b \right ) y^{\prime }&=-4 a \,x^{2} y^{2}-3 b x y+c \,x^{2}+k \\
\end{align*} | [_rational, [_Abel, ‘2nd type‘, ‘class B‘]] | ✗ | ✗ | ✗ | ✗ | 37.644 |
|
| \begin{align*}
\left (y x +a \,x^{n}+b \,x^{2}\right ) y^{\prime }&=y^{2}+c \,x^{n}+b x y \\
\end{align*} |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✓ |
✓ |
✗ |
257.242 |
|
| \begin{align*}
x \left (2 a \,x^{n} y+b \right ) y^{\prime }&=-a \left (3 n +m \right ) x^{n} y^{2}-b \left (2 n +m \right ) y+A \,x^{m}+x \,x^{-n} \\
\end{align*} |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
✓ |
✗ |
27.863 |
|
| \begin{align*}
y^{\prime } y&=-n y^{2}+a \left (2 n +1\right ) {\mathrm e}^{x} y+b y-a^{2} n \,{\mathrm e}^{2 x}-a b \,{\mathrm e}^{x}+c \\
\end{align*} |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
✗ |
✗ |
97.182 |
|
| \begin{align*}
y^{\prime }&=a y^{3}+\frac {b}{x^{{3}/{2}}} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Abel] |
✓ |
✓ |
✓ |
✗ |
8.364 |
|
| \begin{align*}
y^{\prime }&=-y^{3}+3 y a^{2} x^{2}-2 a^{3} x^{3}+a \\
\end{align*} |
[_Abel] |
✗ |
✓ |
✓ |
✗ |
2.249 |
|
| \begin{align*}
y^{\prime }&=-y^{3}+\left (a x +b \right ) y^{2} \\
\end{align*} |
[_Abel] |
✗ |
✓ |
✓ |
✗ |
7.514 |
|
| \begin{align*}
y^{\prime }&=-y^{3}+\frac {y^{2}}{\left (a x +b \right )^{2}} \\
\end{align*} |
[_rational, _Abel] |
✗ |
✓ |
✓ |
✗ |
10.082 |
|
| \begin{align*}
y^{\prime }&=-y^{3}+\frac {y^{2}}{\sqrt {a x +b}} \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
✓ |
✓ |
✓ |
15.069 |
|
| \begin{align*}
y^{\prime }&=a y^{3}+3 a b x y^{2}-b -2 a \,b^{3} x^{3} \\
\end{align*} |
[_Abel] |
✓ |
✓ |
✓ |
✗ |
4.201 |
|
| \begin{align*}
y^{\prime }&=a y^{3} x +b y^{2} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _Abel] |
✓ |
✓ |
✓ |
✓ |
6.846 |
|
| \begin{align*}
y^{\prime }&=a y^{3} x +2 a b \,x^{2} y^{2}-b -2 a \,b^{3} x^{4} \\
\end{align*} |
[_Abel] |
✗ |
✗ |
✗ |
✗ |
5.435 |
|
| \begin{align*}
y^{\prime }&=a \,x^{2 n +1} y^{3}+b \,x^{-n -2} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _Abel] |
✓ |
✓ |
✓ |
✗ |
10.998 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{3}+3 a b \,x^{n +m} y^{2}-b m \,x^{m -1}-2 a \,b^{3} x^{n +3 m} \\
\end{align*} |
[_Abel] |
✓ |
✓ |
✓ |
✗ |
7.417 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{3}+3 a b \,x^{n +m} y^{2}+c \,x^{k} y-2 a \,b^{3} x^{n +3 m}+b c \,x^{m +k}-b m \,x^{m -1} \\
\end{align*} |
[_Abel] |
✓ |
✓ |
✓ |
✗ |
13.229 |
|
| \begin{align*}
9 y^{\prime }&=-x^{m} \left (a \,x^{1-m}+b \right )^{2 \lambda +1} y^{3}-x^{-2 m} \left (9 a +2+9 b m \,x^{m -1}\right ) \left (a \,x^{1-m}+b \right )^{-\lambda -2} \\
\end{align*} |
[_Abel] |
✗ |
✗ |
✗ |
✗ |
13.955 |
|
| \begin{align*}
y^{\prime } x&=a \,x^{4} y^{3}+\left (b \,x^{2}-1\right ) y+c x \\
\end{align*} |
[_rational, _Abel] |
✓ |
✓ |
✓ |
✗ |
5.215 |
|
| \begin{align*}
y^{\prime } x&=a y^{3}+3 a b \,x^{n} y^{2}-b n \,x^{n}-2 a \,b^{3} x^{3 n} \\
\end{align*} |
[_rational, _Abel] |
✓ |
✓ |
✓ |
✗ |
6.759 |
|
| \begin{align*}
y^{\prime } x&=3 x^{2 n +1} y^{3}+\left (b x -n \right ) y+c \,x^{1-n} \\
\end{align*} |
[_rational, _Abel] |
✓ |
✓ |
✓ |
✗ |
6.617 |
|
| \begin{align*}
y^{\prime } x&=a \,x^{n +2} y^{3}+\left (b \,x^{n}-1\right ) y+c \,x^{n -1} \\
\end{align*} | [_rational, _Abel] | ✓ | ✓ | ✓ | ✗ | 7.243 |
|
| \begin{align*}
x^{2} y^{\prime }&=y^{3}-3 a^{2} x^{4} y+2 a^{3} x^{6}+2 a \,x^{3} \\
\end{align*} |
[_rational, _Abel] |
✗ |
✓ |
✓ |
✗ |
2.427 |
|
| \begin{align*}
y^{\prime }&=-\left (a x +b \,x^{m}\right ) y^{3}+y^{2} \\
\end{align*} |
[_Abel] |
✗ |
✗ |
✗ |
✗ |
39.199 |
|
| \begin{align*}
y^{\prime }&=\frac {y^{3}}{\sqrt {a \,x^{2}+b x +c}}+y^{2} \\
\end{align*} |
[_Abel] |
✗ |
✗ |
✗ |
✗ |
56.664 |
|
| \begin{align*}
y^{\prime }&=-y^{3}+a \,{\mathrm e}^{\lambda x} y^{2} \\
\end{align*} |
[_Abel] |
✗ |
✓ |
✓ |
✗ |
5.696 |
|
| \begin{align*}
y^{\prime }&=-y^{3}+3 a^{2} {\mathrm e}^{2 \lambda x} y-2 a^{3} {\mathrm e}^{3 \lambda x}+a \lambda \,{\mathrm e}^{\lambda x} \\
\end{align*} |
[_Abel] |
✗ |
✓ |
✓ |
✗ |
3.332 |
|
| \begin{align*}
y^{\prime }&=-\frac {{\mathrm e}^{2 \lambda x} y^{3}}{3 \lambda }+\frac {2 \lambda ^{2} {\mathrm e}^{-\lambda x}}{3} \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
✓ |
✓ |
✓ |
5.053 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{2 \lambda x} y^{3}+b \,{\mathrm e}^{\lambda x} y^{2}+c y+d \,{\mathrm e}^{-\lambda x} \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
✓ |
✓ |
✗ |
5.546 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{\lambda x} y^{3}+3 a b \,{\mathrm e}^{\lambda x} y^{2}+c y-2 a \,b^{3} {\mathrm e}^{\lambda x}+b c \\
\end{align*} |
[_Abel] |
✓ |
✓ |
✓ |
✗ |
6.297 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{\lambda x} y^{3}+3 a b \,{\mathrm e}^{x \left (\lambda +\mu \right )} y^{2}-2 a \,b^{3} {\mathrm e}^{\left (\lambda +3 \mu \right ) x}-{\mathrm e}^{\mu x} b \mu \\
\end{align*} |
[_Abel] |
✓ |
✓ |
✓ |
✗ |
5.313 |
|
| \begin{align*}
y^{\prime \prime }+a y&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✓ |
2.173 |
|
| \begin{align*}
y^{\prime \prime }-\left (a x +b \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
0.138 |
|
| \begin{align*}
y^{\prime \prime }-\left (a^{2} x^{2}+a \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.662 |
|
| \begin{align*}
y^{\prime \prime }-\left (a \,x^{2}+b \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
1.087 |
|
| \begin{align*}
y^{\prime \prime }+a^{3} x \left (-a x +2\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.506 |
|
| \begin{align*}
y^{\prime \prime }-\left (a \,x^{2}+b x c \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
1.084 |
|
| \begin{align*}
y^{\prime \prime }-a \,x^{n} y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✗ |
0.219 |
|
| \begin{align*}
y^{\prime \prime }-a \left (a \,x^{2 n}+n \,x^{n -1}\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
1.097 |
|
| \begin{align*}
y^{\prime \prime }-a \,x^{-2+n} \left (a \,x^{n}+n +1\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
1.095 |
|
| \begin{align*}
y^{\prime \prime }+\left (a \,x^{2 n}+b \,x^{n -1}\right ) y&=0 \\
\end{align*} | [[_2nd_order, _with_linear_symmetries]] | ✗ | ✓ | ✓ | ✗ | 1.074 |
|
| \begin{align*}
b y+a y^{\prime }+y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✓ |
0.649 |
|
| \begin{align*}
y^{\prime \prime }+a y^{\prime }+\left (b x +c \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.214 |
|
| \begin{align*}
y^{\prime \prime }+a y^{\prime }-\left (b \,x^{2}+c \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
1.810 |
|
| \begin{align*}
y^{\prime \prime }+a y^{\prime }+b \left (-b \,x^{2}+a x +1\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✗ |
0.640 |
|
| \begin{align*}
y^{\prime \prime }+a y^{\prime }+b x \left (-b \,x^{3}+a x +2\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✗ |
0.689 |
|
| \begin{align*}
y^{\prime \prime }+a y^{\prime }+b \left (-b \,x^{2 n}+a \,x^{n}+n \,x^{n -1}\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✗ |
✓ |
✗ |
✗ |
2.697 |
|
| \begin{align*}
y^{\prime \prime }+a y^{\prime }+b \left (-b \,x^{2 n}-a \,x^{n}+n \,x^{n -1}\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
2.635 |
|
| \begin{align*}
y^{\prime \prime }+y^{\prime } x +\left (n -1\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
1.571 |
|
| \begin{align*}
2 n y-2 y^{\prime } x +y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
1.542 |
|
| \begin{align*}
b y+a x y^{\prime }+y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
1.640 |
|
| \begin{align*}
y^{\prime \prime }+a x y^{\prime }+b x y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
1.418 |
|
| \begin{align*}
y^{\prime \prime }+a x y^{\prime }+\left (b x +c \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
1.567 |
|
| \begin{align*}
y^{\prime \prime }+2 a x y^{\prime }+\left (b \,x^{4}+a^{2} x^{2}+c x +a \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
3.258 |
|
| \begin{align*}
y^{\prime \prime }+\left (a x +b \right ) y^{\prime }-a y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.973 |
|
| \begin{align*}
y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+a y&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✗ |
0.939 |
|
| \begin{align*}
y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (a x +b -c \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.639 |
|
| \begin{align*}
y^{\prime \prime }+\left (a x +2 b \right ) y^{\prime }+\left (a b x +b^{2}-a \right ) y&=0 \\
\end{align*} | [[_2nd_order, _with_linear_symmetries]] | ✓ | ✓ | ✓ | ✗ | 0.836 |
|
| \begin{align*}
y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (c x +d \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
2.587 |
|
| \begin{align*}
y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{2}+b x +1\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
3.552 |
|
| \begin{align*}
y^{\prime \prime }+2 \left (a x +b \right ) y^{\prime }+\left (a^{2} x^{2}+2 a b x +c \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.973 |
|
| \begin{align*}
y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
3.918 |
|
| \begin{align*}
y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (-c \,x^{2 n}+a \,x^{n +1}+b \,x^{n}+n \,x^{n -1}\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
1.085 |
|
| \begin{align*}
y^{\prime \prime }+a \left (-b^{2}+x^{2}\right ) y^{\prime }-a \left (x +b \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.871 |
|
| \begin{align*}
y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c \left (a \,x^{2}+b -c \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.717 |
|
| \begin{align*}
y^{\prime \prime }+\left (a \,x^{2}+2 b \right ) y^{\prime }+\left (b \,x^{2} a -a x +b^{2}\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.076 |
|
| \begin{align*}
y^{\prime \prime }+\left (2 x^{2}+a \right ) y^{\prime }+\left (x^{4}+a \,x^{2}+b +2 x \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.863 |
|
| \begin{align*}
y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
3.901 |
|
| \begin{align*}
y^{\prime \prime }+\left (b \,x^{2} a +b x +2 a \right ) y^{\prime }+a^{2} \left (b \,x^{2}+1\right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.895 |
|
| \begin{align*}
y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y^{\prime }+x \left (b \,x^{2} a +b c +2 a \right ) y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.777 |
|