| # |
ODE |
CAS classification |
Solved? |
Maple |
Mma |
Sympy |
time(sec) |
| \begin{align*}
y^{\prime \prime }&=a +b x +c y^{2} \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
0.869 |
|
| \begin{align*}
y^{\prime \prime }&=2 y^{3} \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✓ |
34.771 |
|
| \begin{align*}
y^{\prime \prime }&=a +b y+2 y^{3} \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✗ |
205.765 |
|
| \begin{align*}
y^{\prime \prime }&=a +y x +2 y^{3} \\
\end{align*} |
[[_Painleve, ‘2nd‘]] |
✗ |
✗ |
✗ |
✗ |
0.784 |
|
| \begin{align*}
y^{\prime \prime }&=f \left (x \right )+g \left (x \right ) y+2 y^{3} \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
1.319 |
|
| \begin{align*}
y^{\prime \prime }&=a -2 a b x y+2 y^{3} b^{2} \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
0.690 |
|
| \begin{align*}
y^{\prime \prime }&=\operatorname {a0} +\operatorname {a2} y+\operatorname {a1} x y+\operatorname {a3} y^{3} \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
0.915 |
|
| \begin{align*}
y^{\prime \prime }&=\operatorname {a0} +\operatorname {a1} y+\operatorname {a2} y^{2}+\operatorname {a3} y^{3} \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✗ |
192.496 |
|
| \begin{align*}
a \,x^{r} y^{s}+y^{\prime \prime }&=0 \\
\end{align*} |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
0.996 |
|
| \begin{align*}
a \sin \left (y\right )+y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✗ |
46.217 |
|
| \begin{align*}
a \,{\mathrm e}^{y}+y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✓ |
58.371 |
|
| \begin{align*}
y^{\prime \prime }&=f \left (y\right ) \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✗ |
9.778 |
|
| \begin{align*}
y \left (2 f \left (x \right )^{2}+f^{\prime }\left (x \right )\right )+3 f \left (x \right ) y^{\prime }+y^{\prime \prime }&=2 y^{3} \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
3.532 |
|
| \begin{align*}
y y^{\prime }+y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
3.274 |
|
| \begin{align*}
y y^{\prime }+y^{\prime \prime }&=y^{3} \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✗ |
206.774 |
|
| \begin{align*}
a y+y y^{\prime }+y^{\prime \prime }&=y^{3} \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✗ |
✓ |
✓ |
✗ |
395.638 |
|
| \begin{align*}
y y^{\prime }+y^{\prime \prime }&=-12 f \left (x \right ) y+y^{3}+12 f^{\prime }\left (x \right ) \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
3.338 |
|
| \begin{align*}
2 a^{2} y+a y^{2}+\left (3 a +y\right ) y^{\prime }+y^{\prime \prime }&=y^{3} \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✗ |
✓ |
✓ |
✗ |
141.396 |
|
| \begin{align*}
y^{\prime \prime }&=f \left (x \right ) y^{2}+y^{3}+y \left (-2 f \left (x \right )^{2}+f^{\prime }\left (x \right )\right )+\left (3 f \left (x \right )-y\right ) y^{\prime } \\
\end{align*} |
[NONE] |
✗ |
✓ |
✗ |
✗ |
4.448 |
|
| \begin{align*}
y^{\prime \prime }&=\operatorname {f2} \left (x \right )+\operatorname {f3} \left (x \right ) y+\operatorname {f1} \left (x \right ) y^{2}+y^{3}+\left (3 \operatorname {f1} \left (x \right )-y\right ) y^{\prime } \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
4.914 |
|
| \begin{align*}
y^{\prime \prime }&=\operatorname {g3} \left (x \right )+\operatorname {g2} \left (x \right ) y+\operatorname {g1} \left (x \right ) y^{2}+\operatorname {g0} \left (x \right ) y^{3}+\left (\operatorname {f1} \left (x \right )+\operatorname {f0} \left (x \right ) y\right ) y^{\prime } \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
5.718 |
|
| \begin{align*}
y^{\prime \prime }&=f^{\prime }\left (x \right ) y+\left (f \left (x \right )-2 y\right ) y^{\prime } \\
\end{align*} |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✗ |
✗ |
✗ |
20.873 |
|
| \begin{align*}
y^{\prime \prime }&=g \left (x \right )+f \left (x \right ) y^{2}+\left (f \left (x \right )-2 y\right ) y^{\prime } \\
\end{align*} |
[[_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✗ |
✗ |
✗ |
3.539 |
|
| \begin{align*}
y^{\prime \prime }&=\operatorname {f3} \left (x \right )+\operatorname {f2} \left (x \right ) y^{2}+\left (\operatorname {f1} \left (x \right )-2 y\right ) y^{\prime } \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
3.766 |
|
| \begin{align*}
y^{\prime \prime }&=\operatorname {f4} \left (x \right )+\operatorname {f3} \left (x \right ) y+\operatorname {f2} \left (x \right ) y^{2}+\left (\operatorname {f1} \left (x \right )-2 y\right ) y^{\prime } \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
4.346 |
|
| \begin{align*}
y^{\prime \prime }&=a +4 y b^{2}+3 b y^{2}+3 y y^{\prime } \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
114.726 |
|
| \begin{align*}
3 y y^{\prime }+y^{\prime \prime }&=f \left (x \right )+g \left (x \right ) y-y^{3} \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
3.815 |
|
| \begin{align*}
y^{\prime \prime }&=f \left (x \right ) y^{2}-y^{3}+\left (f \left (x \right )-3 y\right ) y^{\prime } \\
\end{align*} |
[[_2nd_order, _with_potential_symmetries]] |
✗ |
✓ |
✗ |
✗ |
3.244 |
|
| \begin{align*}
y^{\prime \prime }&=a \left (1+2 y y^{\prime }\right ) \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
239.062 |
|
| \begin{align*}
b y+a \left (y^{2}-1\right ) y^{\prime }+y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
✗ |
87.363 |
|
| \begin{align*}
g \left (x , y\right )+f \left (x , y\right ) y^{\prime }+y^{\prime \prime }&=0 \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
0.963 |
|
| \begin{align*}
y^{\prime \prime }&=2 x +\left (x^{2}-y^{\prime }\right )^{2} \\
\end{align*} |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✓ |
5.421 |
|
| \begin{align*}
2 \cot \left (x \right ) y^{\prime }+2 \tan \left (y\right ) {y^{\prime }}^{2}+y^{\prime \prime }&=0 \\
\end{align*} |
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
1.293 |
|
| \begin{align*}
y^{\prime \prime }&=a {y^{\prime }}^{2} \\
\end{align*} |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✓ |
4.870 |
|
| \begin{align*}
y^{\prime \prime }&=a^{2}+b^{2} {y^{\prime }}^{2} \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✓ |
37.063 |
|
| \begin{align*}
b y+a {y^{\prime }}^{2}+y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✗ |
8.931 |
|
| \begin{align*}
b \sin \left (y\right )+a {y^{\prime }}^{2}+y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✗ |
38.327 |
|
| \begin{align*}
c y+b y^{\prime }+a {y^{\prime }}^{2}+y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
✗ |
161.638 |
|
| \begin{align*}
y^{\prime \prime }&={\mathrm e}^{x} {y^{\prime }}^{2} \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
2.017 |
|
| \begin{align*}
f \left (x \right ) y^{\prime }+g \left (x \right ) {y^{\prime }}^{2}+y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.233 |
|
| \begin{align*}
b y+a y {y^{\prime }}^{2}+y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✗ |
16.451 |
|
| \begin{align*}
g \left (y\right )+f \left (y\right ) {y^{\prime }}^{2}+y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✗ |
15.211 |
|
| \begin{align*}
f \left (x \right ) y^{\prime }+g \left (y\right ) {y^{\prime }}^{2}+y^{\prime \prime }&=0 \\
\end{align*} |
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
0.717 |
|
| \begin{align*}
f \left (y\right ) y^{\prime }+g \left (y\right ) {y^{\prime }}^{2}+y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
2.678 |
|
| \begin{align*}
h \left (y\right )+f \left (y\right ) y^{\prime }+g \left (y\right ) {y^{\prime }}^{2}+y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
✗ |
19.116 |
|
| \begin{align*}
y^{\prime }+{y^{\prime }}^{3}+y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✓ |
11.190 |
|
| \begin{align*}
y^{\prime \prime }&=\left (-x +a \right ) {y^{\prime }}^{3} \\
\end{align*} |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
✓ |
✓ |
9.800 |
|
| \begin{align*}
\left ({\mathrm e}^{2 y}+x \right ) {y^{\prime }}^{3}+y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_exponential_symmetries], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
✗ |
✗ |
2.776 |
|
| \begin{align*}
2 y^{\prime }+4 {y^{\prime }}^{3}+y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✓ |
11.232 |
|
| \begin{align*}
a {y^{\prime }}^{3}+y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
✓ |
✓ |
13.951 |
|
| \begin{align*}
y^{\prime \prime }&=x {y^{\prime }}^{3} \\
\end{align*} |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
✓ |
✓ |
7.092 |
|
| \begin{align*}
\left (a x +b y\right ) {y^{\prime }}^{3}+y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_exponential_symmetries], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
✓ |
✗ |
3.757 |
|
| \begin{align*}
a y \left (1+{y^{\prime }}^{2}\right )^{2}+y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✗ |
21.628 |
|
| \begin{align*}
y^{\prime \prime }&=a \left (x y^{\prime }-y\right )^{k} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
2.025 |
|
| \begin{align*}
g \left (x \right ) y^{\prime }+f \left (x \right ) {y^{\prime }}^{k}+y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✗ |
5.260 |
|
| \begin{align*}
y^{\prime \prime }&=A \,x^{a} y^{b} {y^{\prime }}^{c} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
1.354 |
|
| \begin{align*}
y^{\prime \prime }&=a \sqrt {1+{y^{\prime }}^{2}} \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✗ |
24.566 |
|
| \begin{align*}
y^{\prime \prime }&=a \sqrt {b y^{2}+{y^{\prime }}^{2}} \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✗ |
✓ |
✓ |
✗ |
372.311 |
|
| \begin{align*}
y^{\prime \prime }&=a \left (1+{y^{\prime }}^{2}\right )^{{3}/{2}} \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✗ |
39.098 |
|
| \begin{align*}
y^{\prime \prime }&=a x \left (1+{y^{\prime }}^{2}\right )^{{3}/{2}} \\
\end{align*} |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
✓ |
✗ |
4.369 |
|
| \begin{align*}
y^{\prime \prime }&=a y \left (1+{y^{\prime }}^{2}\right )^{{3}/{2}} \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✗ |
26.001 |
|
| \begin{align*}
y^{\prime \prime }&=a y {\left (1+\left (b -y^{\prime }\right )^{2}\right )}^{{3}/{2}} \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✗ |
51.510 |
|
| \begin{align*}
y^{\prime \prime }&=a \left (b +c x +y\right ) \left (1+{y^{\prime }}^{2}\right )^{{3}/{2}} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
✓ |
✓ |
✗ |
2.003 |
|
| \begin{align*}
y^{3} y^{\prime }+y^{\prime \prime }&=y y^{\prime } \sqrt {y^{4}+4 y^{\prime }} \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✗ |
330.502 |
|
| \begin{align*}
y^{\prime \prime }&=f \left (y^{\prime }\right ) \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✗ |
2.663 |
|
| \begin{align*}
y^{\prime \prime }&=f \left (a x +b y, y^{\prime }\right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
1.435 |
|
| \begin{align*}
y^{\prime \prime }&=f \left (x , \frac {y^{\prime }}{y}\right ) y \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
0.789 |
|
| \begin{align*}
y^{\prime \prime }&=x^{-2+n} f \left (y x^{-n}, x^{1-n} y^{\prime }\right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
1.476 |
|
| \begin{align*}
2 y^{\prime \prime }&=1+12 y^{2} \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✗ |
62.012 |
|
| \begin{align*}
2 y^{\prime \prime }&=y \left (a -y^{2}\right ) \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✗ |
60.175 |
|
| \begin{align*}
9 {y^{\prime }}^{4}+8 y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✓ |
14.168 |
|
| \begin{align*}
a \,{\mathrm e}^{y} x +y^{\prime }+x y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1]] |
✗ |
✓ |
✓ |
✗ |
1.382 |
|
| \begin{align*}
y^{5} x +2 y^{\prime }+x y^{\prime \prime }&=0 \\
\end{align*} |
[_Emden, [_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
0.997 |
|
| \begin{align*}
x y^{n}+2 y^{\prime }+x y^{\prime \prime }&=0 \\
\end{align*} |
[_Emden, [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
1.118 |
|
| \begin{align*}
x^{m} y^{n}+2 y^{\prime }+x y^{\prime \prime }&=0 \\
\end{align*} |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
1.180 |
|
| \begin{align*}
a \,x^{m} y^{n}+2 y^{\prime }+x y^{\prime \prime }&=0 \\
\end{align*} |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
1.144 |
|
| \begin{align*}
b \,{\mathrm e}^{y} x +a y^{\prime }+x y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
1.092 |
|
| \begin{align*}
\left (-a \,x^{2}+2\right ) y^{\prime }+x y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
2.787 |
|
| \begin{align*}
x y^{\prime \prime }&=\left (1-y\right ) y^{\prime } \\
\end{align*} |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
5.644 |
|
| \begin{align*}
{y^{\prime }}^{2} x +x y^{\prime \prime }&=y^{\prime } \\
\end{align*} |
[[_2nd_order, _missing_y], _Liouville, [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✓ |
2.230 |
|
| \begin{align*}
x y^{\prime \prime }&={y^{\prime }}^{2} x +y^{\prime } \\
\end{align*} |
[[_2nd_order, _missing_y], _Liouville, [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✓ |
2.208 |
|
| \begin{align*}
-2 y^{\prime }+2 {y^{\prime }}^{2} x +x y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y], _Liouville, [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✓ |
2.197 |
|
| \begin{align*}
x y^{\prime \prime }&=-y^{2}-2 y^{\prime }+{y^{\prime }}^{2} x^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
1.138 |
|
| \begin{align*}
2 y^{\prime }+a \,x^{2} {y^{\prime }}^{2}+x y^{\prime \prime }&=b \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✗ |
6.421 |
|
| \begin{align*}
\left (-y+a x y^{\prime }\right )^{2}+x y^{\prime \prime }&=b \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
1.101 |
|
| \begin{align*}
x y^{\prime \prime }&={y^{\prime }}^{3}+y^{\prime } \\
\end{align*} |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
✓ |
✓ |
2.342 |
|
| \begin{align*}
2 y^{\prime }+x y^{\prime \prime }&=a \,x^{2 k} {y^{\prime }}^{k} \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✗ |
2.099 |
|
| \begin{align*}
y^{\prime }+{y^{\prime }}^{3}+2 x y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
✓ |
✓ |
4.077 |
|
| \begin{align*}
a y \left (1-y^{n}\right )+x^{2} y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
1.019 |
|
| \begin{align*}
a \,{\mathrm e}^{-1+y}+x^{2} y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
1.017 |
|
| \begin{align*}
\left (a +1\right ) x y^{\prime }+x^{2} y^{\prime \prime }&=x^{k} f \left (x^{k} y, k y+x y^{\prime }\right ) \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
2.902 |
|
| \begin{align*}
{y^{\prime }}^{2}+x^{2} y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
✓ |
✓ |
1.808 |
|
| \begin{align*}
x^{2} y^{\prime \prime }&=\left (3 x -2 y^{\prime }\right ) y^{\prime } \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.690 |
|
| \begin{align*}
2+4 x y^{\prime }+{y^{\prime }}^{2} x^{2}+x^{2} y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✓ |
1.832 |
|
| \begin{align*}
x^{2} y^{\prime \prime }&=6 y-4 x^{2} y^{2}+x^{4} {y^{\prime }}^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
2.250 |
|
| \begin{align*}
a \left (x y^{\prime }-y\right )^{2}+x^{2} y^{\prime \prime }&=b \,x^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
1.395 |
|
| \begin{align*}
2 y x +a \,x^{4} {y^{\prime }}^{2}+x^{2} y^{\prime \prime }&=b \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
1.211 |
|
| \begin{align*}
b x +a y {y^{\prime }}^{2}+x^{2} y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
0.743 |
|
| \begin{align*}
x^{2} y^{\prime \prime }&=\sqrt {b y^{2}+a \,x^{2} {y^{\prime }}^{2}} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
2.558 |
|
| \begin{align*}
x^{2} y^{\prime \prime }&=f \left (\frac {x y^{\prime }}{y}\right ) y \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
36.069 |
|