| # |
ODE |
CAS classification |
Solved? |
Maple |
Mma |
Sympy |
time(sec) |
| \begin{align*}
b x +a y {y^{\prime }}^{2}+x^{2} y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
0.487 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-\sqrt {b y^{2}+a \,x^{2} {y^{\prime }}^{2}}&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
1.195 |
|
| \begin{align*}
\left (x^{2}+1\right ) y^{\prime \prime }+1+{y^{\prime }}^{2}&=0 \\
\end{align*} |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
✓ |
✓ |
0.378 |
|
| \begin{align*}
4 x^{2} y^{\prime \prime }-x^{4} {y^{\prime }}^{2}+4 y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
0.648 |
|
| \begin{align*}
2 y+a y^{3}+9 x^{2} y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
0.490 |
|
| \begin{align*}
24+12 y x +x^{3} \left (-y^{3}+y y^{\prime }+y^{\prime \prime }\right )&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✓ |
✗ |
1.312 |
|
| \begin{align*}
x^{3} y^{\prime \prime }-a \left (-y+y^{\prime } x \right )^{2}&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
0.937 |
|
| \begin{align*}
2 x^{3} y^{\prime \prime }+x^{2} \left (9+2 y x \right ) y^{\prime }+b +x y \left (a +3 y x -2 y^{2} x^{2}\right )&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
1.557 |
|
| \begin{align*}
2 \left (-x^{k}+4 x^{3}\right ) \left (-y^{3}+y y^{\prime }+y^{\prime \prime }\right )-\left (-12 x^{2}+k \,x^{-1+k}\right ) \left (y^{2}+3 y^{\prime }\right )+a x y+b&=0 \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
3.419 |
|
| \begin{align*}
x^{4} y^{\prime \prime }+a^{2} y^{n}&=0 \\
\end{align*} |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
0.535 |
|
| \begin{align*}
x^{4} y^{\prime \prime }-x \left (x^{2}+2 y\right ) y^{\prime }+4 y^{2}&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
0.998 |
|
| \begin{align*}
x^{4} y^{\prime \prime }-x^{2} y^{\prime } \left (x +y^{\prime }\right )+4 y^{2}&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
1.058 |
|
| \begin{align*}
\left (-y+y^{\prime } x \right )^{3}+x^{4} y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
1.751 |
|
| \begin{align*}
\sqrt {x}\, y^{\prime \prime }-y^{{3}/{2}}&=0 \\
\end{align*} |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
0.436 |
|
| \begin{align*}
\left (a \,x^{2}+b x +c \right )^{{3}/{2}} y^{\prime \prime }-F \left (\frac {y}{\sqrt {a \,x^{2}+b x +c}}\right )&=0 \\
\end{align*} |
[NONE] |
✗ |
✓ |
✓ |
✗ |
51.047 |
|
| \begin{align*}
y y^{\prime \prime }-a&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✗ |
0.996 |
|
| \begin{align*}
y y^{\prime \prime }-a x&=0 \\
\end{align*} |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
0.368 |
|
| \begin{align*}
y y^{\prime \prime }-a \,x^{2}&=0 \\
\end{align*} |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
0.408 |
|
| \begin{align*}
y y^{\prime \prime }+{y^{\prime }}^{2}-a&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
2.609 |
|
| \begin{align*}
y y^{\prime \prime }+y^{2}-a x -b&=0 \\
\end{align*} |
[NONE] |
✗ |
✗ |
✓ |
✗ |
0.394 |
|
| \begin{align*}
y y^{\prime \prime }+{y^{\prime }}^{2}-y^{\prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
1.052 |
|
| \begin{align*}
y y^{\prime \prime }-{y^{\prime }}^{2}+1&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✗ |
2.624 |
|
| \begin{align*}
y y^{\prime \prime }-{y^{\prime }}^{2}-1&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✗ |
2.222 |
|
| \begin{align*}
y y^{\prime \prime }-{y^{\prime }}^{2}+{\mathrm e}^{x} y \left (c y^{2}+d \right )+{\mathrm e}^{2 x} \left (b +a y^{4}\right )&=0 \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
0.696 |
|
| \begin{align*}
y y^{\prime \prime }-{y^{\prime }}^{2}-\ln \left (y\right ) y^{2}&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
2.224 |
|
| \begin{align*}
y y^{\prime \prime }-{y^{\prime }}^{2}+f \left (x \right ) y^{\prime }-y f^{\prime }\left (x \right )-y^{3}&=0 \\
\end{align*} |
[NONE] |
✗ |
✗ |
✓ |
✗ |
1.605 |
|
| \begin{align*}
y y^{\prime \prime }-{y^{\prime }}^{2}+f^{\prime }\left (x \right ) y^{\prime }-y f^{\prime \prime }\left (x \right )+f \left (x \right ) y^{3}-y^{4}&=0 \\
\end{align*} |
[NONE] |
✗ |
✗ |
✓ |
✗ |
0.950 |
|
| \begin{align*}
y y^{\prime \prime }-{y^{\prime }}^{2}+a y y^{\prime }+b y^{2}&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
10.737 |
|
| \begin{align*}
y y^{\prime \prime }-{y^{\prime }}^{2}+a y y^{\prime }-2 a y^{2}+y^{3} b&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
✗ |
34.158 |
|
| \begin{align*}
y y^{\prime \prime }-{y^{\prime }}^{2}-\left (a y-1\right ) y^{\prime }+2 y^{2} a^{2}-2 b^{2} y^{3}+a y&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✓ |
✗ |
79.589 |
|
| \begin{align*}
y y^{\prime \prime }-{y^{\prime }}^{2}+\left (a y-1\right ) y^{\prime }-y \left (1+y\right ) \left (b^{2} y^{2}-a^{2}\right )&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
✗ |
119.023 |
|
| \begin{align*}
y y^{\prime \prime }-{y^{\prime }}^{2}+\left (\tan \left (x \right )+\cot \left (x \right )\right ) y y^{\prime }+\left (\cos \left (x \right )^{2}-n^{2} \cot \left (x \right )^{2}\right ) y^{2} \ln \left (y\right )&=0 \\
\end{align*} |
[[_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
5.393 |
|
| \begin{align*}
y y^{\prime \prime }-{y^{\prime }}^{2}-f \left (x \right ) y y^{\prime }-g \left (x \right ) y^{2}&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
2.393 |
|
| \begin{align*}
y y^{\prime \prime }-{y^{\prime }}^{2}+\left (g \left (x \right )+f \left (x \right ) y^{2}\right ) y^{\prime }-y \left (g^{\prime }\left (x \right )-f^{\prime }\left (x \right ) y^{2}\right )&=0 \\
\end{align*} |
[[_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✗ |
✓ |
✗ |
3.000 |
|
| \begin{align*}
y y^{\prime \prime }-3 {y^{\prime }}^{2}+3 y y^{\prime }-y^{2}&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
22.712 |
|
| \begin{align*}
y y^{\prime \prime }-a {y^{\prime }}^{2}&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
0.511 |
|
| \begin{align*}
y y^{\prime \prime }+a \left (1+{y^{\prime }}^{2}\right )&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✗ |
1.896 |
|
| \begin{align*}
y y^{\prime \prime }+a {y^{\prime }}^{2}+y^{3} b&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✗ |
2.205 |
|
| \begin{align*}
y y^{\prime \prime }+a {y^{\prime }}^{2}+b y y^{\prime }+c y^{2}+d y^{1-a}&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
363.589 |
|
| \begin{align*}
y y^{\prime \prime }+a {y^{\prime }}^{2}+b y^{2} y^{\prime }+c y^{4}&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✗ |
78.965 |
|
| \begin{align*}
y y^{\prime \prime }-\frac {\left (-1+a \right ) {y^{\prime }}^{2}}{a}-f \left (x \right ) y^{2} y^{\prime }+\frac {a f \left (x \right )^{2} y^{4}}{\left (2+a \right )^{2}}-\frac {a f^{\prime }\left (x \right ) y^{3}}{2+a}&=0 \\
\end{align*} |
[NONE] |
✗ |
✗ |
✓ |
✗ |
2.945 |
|
| \begin{align*}
y y^{\prime \prime }-{y^{\prime }}^{2}-1-2 a y \left (1+{y^{\prime }}^{2}\right )^{{3}/{2}}&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✗ |
✓ |
✓ |
✗ |
366.648 |
|
| \begin{align*}
-y^{\prime }+{y^{\prime }}^{2}+\left (x +y\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
0.395 |
|
| \begin{align*}
2 y^{\prime } \left (1+y^{\prime }\right )+\left (x -y\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
✓ |
✓ |
✗ |
1.291 |
|
| \begin{align*}
\left (x -y\right ) y^{\prime \prime }-\left (1+y^{\prime }\right ) \left (1+{y^{\prime }}^{2}\right )&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
✓ |
✓ |
✗ |
1.077 |
|
| \begin{align*}
\left (x -y\right ) y^{\prime \prime }-h \left (y^{\prime }\right )&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
✓ |
✓ |
✗ |
3.803 |
|
| \begin{align*}
1+{y^{\prime }}^{2}+2 y y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✗ |
2.055 |
|
| \begin{align*}
2 y y^{\prime \prime }-{y^{\prime }}^{2}+a&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✗ |
1.782 |
|
| \begin{align*}
2 y y^{\prime \prime }-{y^{\prime }}^{2}+f \left (x \right ) y^{2}+a&=0 \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
0.969 |
|
| \begin{align*}
2 y y^{\prime \prime }-{y^{\prime }}^{2}-8 y^{3}&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✗ |
137.177 |
|
| \begin{align*}
2 y y^{\prime \prime }-{y^{\prime }}^{2}-8 y^{3}-4 y^{2}&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✗ |
1.286 |
|
| \begin{align*}
2 y y^{\prime \prime }-{y^{\prime }}^{2}-4 y^{2} \left (x +2 y\right )&=0 \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
0.495 |
|
| \begin{align*}
2 y y^{\prime \prime }-{y^{\prime }}^{2}+\left (a y+b \right ) y^{2}&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✗ |
1.425 |
|
| \begin{align*}
2 y y^{\prime \prime }-{y^{\prime }}^{2}+1+2 x y^{2}+a y^{3}&=0 \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
0.494 |
|
| \begin{align*}
2 y y^{\prime \prime }-{y^{\prime }}^{2}+\left (a y+b x \right ) y^{2}&=0 \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
0.457 |
|
| \begin{align*}
2 y y^{\prime \prime }-{y^{\prime }}^{2}-3 y^{4}&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✗ |
1.380 |
|
| \begin{align*}
2 y y^{\prime \prime }-{y^{\prime }}^{2}+b -4 \left (x^{2}+a \right ) y^{2}-8 x y^{3}-3 y^{4}&=0 \\
\end{align*} |
[[_Painleve, ‘4th‘]] |
✗ |
✗ |
✗ |
✗ |
0.603 |
|
| \begin{align*}
2 y y^{\prime \prime }-3 {y^{\prime }}^{2}&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
0.482 |
|
| \begin{align*}
2 y y^{\prime \prime }-3 {y^{\prime }}^{2}-4 y^{2}&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
2.181 |
|
| \begin{align*}
2 y y^{\prime \prime }-3 {y^{\prime }}^{2}+f \left (x \right ) y^{2}&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
0.837 |
|
| \begin{align*}
2 y y^{\prime \prime }-6 {y^{\prime }}^{2}+y^{2} \left (1+a y^{3}\right )&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✗ |
40.745 |
|
| \begin{align*}
2 y y^{\prime \prime }-{y^{\prime }}^{2} \left (1+{y^{\prime }}^{2}\right )&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✗ |
0.619 |
|
| \begin{align*}
2 \left (y-a \right ) y^{\prime \prime }+{y^{\prime }}^{2}+1&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✗ |
2.192 |
|
| \begin{align*}
3 y y^{\prime \prime }-2 {y^{\prime }}^{2}-a \,x^{2}-b x -c&=0 \\
\end{align*} |
[NONE] |
✗ |
✓ |
✓ |
✗ |
0.592 |
|
| \begin{align*}
3 y y^{\prime \prime }-5 {y^{\prime }}^{2}&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
0.928 |
|
| \begin{align*}
4 y y^{\prime \prime }-3 {y^{\prime }}^{2}+4 y&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✗ |
2.333 |
|
| \begin{align*}
4 y y^{\prime \prime }-3 {y^{\prime }}^{2}-12 y^{3}&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✗ |
2.334 |
|
| \begin{align*}
4 y y^{\prime \prime }-3 {y^{\prime }}^{2}+a y^{3}+b y^{2}+c y&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✗ |
1.982 |
|
| \begin{align*}
4 y y^{\prime \prime }-5 {y^{\prime }}^{2}+a y^{2}&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
1.737 |
|
| \begin{align*}
12 y y^{\prime \prime }-15 {y^{\prime }}^{2}+8 y^{3}&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✗ |
3.452 |
|
| \begin{align*}
n y y^{\prime \prime }-\left (n -1\right ) {y^{\prime }}^{2}&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
0.560 |
|
| \begin{align*}
a y y^{\prime \prime }+b {y^{\prime }}^{2}+\operatorname {c4} y^{4}+\operatorname {c3} y^{3}+\operatorname {c2} y^{2}+\operatorname {c1} y+\operatorname {c0}&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✗ |
4.064 |
|
| \begin{align*}
a y y^{\prime \prime }+b {y^{\prime }}^{2}-\frac {y y^{\prime }}{\sqrt {c^{2}+x^{2}}}&=0 \\
\end{align*} |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
0.409 |
|
| \begin{align*}
\left (a y+b \right ) y^{\prime \prime }+c {y^{\prime }}^{2}&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
0.929 |
|
| \begin{align*}
x y y^{\prime \prime }+x {y^{\prime }}^{2}-y y^{\prime }&=0 \\
\end{align*} |
[[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
0.996 |
|
| \begin{align*}
f \left (x \right )+a y y^{\prime }+x {y^{\prime }}^{2}+x y y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
2.007 |
|
| \begin{align*}
x y y^{\prime \prime }-x {y^{\prime }}^{2}+y y^{\prime }+x \left (d +a y^{4}\right )+y \left (c +b y^{2}\right )&=0 \\
\end{align*} |
[[_Painleve, ‘3rd‘]] |
✗ |
✗ |
✗ |
✗ |
1.105 |
|
| \begin{align*}
x y y^{\prime \prime }-x {y^{\prime }}^{2}+a y y^{\prime }+y^{3} b x&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
0.967 |
|
| \begin{align*}
a y y^{\prime }+2 x {y^{\prime }}^{2}+x y y^{\prime \prime }&=0 \\
\end{align*} |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
0.957 |
|
| \begin{align*}
x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+\left (1+y\right ) y^{\prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
1.039 |
|
| \begin{align*}
a y y^{\prime }-2 x {y^{\prime }}^{2}+x y y^{\prime \prime }&=0 \\
\end{align*} |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
0.220 |
|
| \begin{align*}
4 y y^{\prime }-4 x {y^{\prime }}^{2}+x y y^{\prime \prime }&=0 \\
\end{align*} |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
0.788 |
|
| \begin{align*}
x y y^{\prime \prime }+\left (\frac {a x}{\sqrt {b^{2}-x^{2}}}-x \right ) {y^{\prime }}^{2}-y y^{\prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
1.839 |
|
| \begin{align*}
x \left (x +y\right ) y^{\prime \prime }+x {y^{\prime }}^{2}+\left (x -y\right ) y^{\prime }-y&=0 \\
\end{align*} |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
0.546 |
|
| \begin{align*}
2 x y y^{\prime \prime }-x {y^{\prime }}^{2}+y y^{\prime }&=0 \\
\end{align*} |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
0.466 |
|
| \begin{align*}
x^{2} \left (-1+y\right ) y^{\prime \prime }-2 x^{2} {y^{\prime }}^{2}-2 x \left (-1+y\right ) y^{\prime }-2 y \left (-1+y\right )^{2}&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✗ |
✗ |
33.888 |
|
| \begin{align*}
x^{2} \left (x +y\right ) y^{\prime \prime }-\left (-y+y^{\prime } x \right )^{2}&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
10.340 |
|
| \begin{align*}
x^{2} \left (x -y\right ) y^{\prime \prime }+a \left (-y+y^{\prime } x \right )^{2}&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
8.408 |
|
| \begin{align*}
2 x^{2} y y^{\prime \prime }-x^{2} \left (1+{y^{\prime }}^{2}\right )+y^{2}&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
1.378 |
|
| \begin{align*}
a \,x^{2} y y^{\prime \prime }+b \,x^{2} {y^{\prime }}^{2}+c x y y^{\prime }+d y^{2}&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
8.048 |
|
| \begin{align*}
x \left (x +1\right )^{2} y y^{\prime \prime }-x \left (x +1\right )^{2} {y^{\prime }}^{2}+2 \left (x +1\right )^{2} y y^{\prime }-a \left (2+x \right ) y^{2}&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
9.511 |
|
| \begin{align*}
8 \left (-x^{3}+1\right ) y y^{\prime \prime }-4 \left (-x^{3}+1\right ) {y^{\prime }}^{2}-12 x^{2} y y^{\prime }+3 x y^{2}&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
8.595 |
|
| \begin{align*}
y^{2} y^{\prime \prime }-a&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✓ |
52.406 |
|
| \begin{align*}
a x +y {y^{\prime }}^{2}+y^{2} y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
1.060 |
|
| \begin{align*}
y^{2} y^{\prime \prime }+y {y^{\prime }}^{2}-a x -b&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
0.968 |
|
| \begin{align*}
\left (1-2 y\right ) {y^{\prime }}^{2}+\left (1+y^{2}\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
0.711 |
|
| \begin{align*}
\left (1+y^{2}\right ) y^{\prime \prime }-3 y {y^{\prime }}^{2}&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
0.401 |
|
| \begin{align*}
\left (x +y^{2}\right ) y^{\prime \prime }-2 \left (x -y^{2}\right ) {y^{\prime }}^{3}+y^{\prime } \left (1+4 y y^{\prime }\right )&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
✓ |
✗ |
✗ |
9.456 |
|
| \begin{align*}
\left (x^{2}+y^{2}\right ) y^{\prime \prime }-\left (1+{y^{\prime }}^{2}\right ) \left (-y+y^{\prime } x \right )&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
✓ |
✓ |
✗ |
8.790 |
|
| \begin{align*}
\left (x^{2}+y^{2}\right ) y^{\prime \prime }-2 \left (1+{y^{\prime }}^{2}\right ) \left (-y+y^{\prime } x \right )&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
✓ |
✓ |
✗ |
8.701 |
|