|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}y^{\prime \prime }+6 a^{10} y^{11}-y = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
0.994 |
|
|
\[
{}y^{\prime \prime }-\frac {1}{\left (a y^{2}+b x y+c \,x^{2}+\alpha y+\beta x +\gamma \right )^{{3}/{2}}} = 0
\] |
[NONE] |
✗ |
0.302 |
|
|
\[
{}y^{\prime \prime }-{\mathrm e}^{y} = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
35.629 |
|
|
\[
{}y^{\prime \prime }+a \,{\mathrm e}^{x} \sqrt {y} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.174 |
|
|
\[
{}y^{\prime \prime }+{\mathrm e}^{x} \sin \left (y\right ) = 0
\] |
[NONE] |
✗ |
0.244 |
|
|
\[
{}y^{\prime \prime }+a \sin \left (y\right ) = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
2.029 |
|
|
\[
{}y^{\prime \prime }+a^{2} \sin \left (y\right )-\beta \sin \left (x \right ) = 0
\] |
[NONE] |
✗ |
0.644 |
|
|
\[
{}y^{\prime \prime }+a^{2} \sin \left (y\right )-\beta f \left (x \right ) = 0
\] |
[NONE] |
✗ |
0.516 |
|
|
\[
{}y^{\prime \prime } = \frac {f \left (\frac {y}{\sqrt {x}}\right )}{x^{{3}/{2}}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.340 |
|
|
\[
{}y^{\prime \prime }-3 y^{\prime }-y^{2}-2 y = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
1.560 |
|
|
\[
{}y^{\prime \prime }-7 y^{\prime }-y^{{3}/{2}}+12 y = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
5.280 |
|
|
\[
{}y^{\prime \prime }+5 a y^{\prime }-6 y^{2}+6 a^{2} y = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
1.989 |
|
|
\[
{}y^{\prime \prime }+3 a y^{\prime }-2 y^{3}+2 a^{2} y = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
2.303 |
|
|
\[
{}y^{\prime \prime }-\frac {\left (3 n +4\right ) y^{\prime }}{n}-\frac {2 \left (n +1\right ) \left (n +2\right ) y \left (y^{\frac {n}{n +1}}-1\right )}{n^{2}} = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
125.833 |
|
|
\[
{}y^{\prime \prime }+a y^{\prime }+b y^{n}+\frac {\left (a^{2}-1\right ) y}{4} = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
4.382 |
|
|
\[
{}y^{\prime \prime }+a y^{\prime }+b \,x^{v} y^{n} = 0
\] |
[NONE] |
✗ |
0.151 |
|
|
\[
{}y^{\prime \prime }+a y^{\prime }+b \,{\mathrm e}^{y}-2 a = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
3.223 |
|
|
\[
{}y^{\prime \prime }+a y^{\prime }+f \left (x \right ) \sin \left (y\right ) = 0
\] |
[NONE] |
✗ |
0.319 |
|
|
\[
{}y^{\prime \prime }+y y^{\prime }-y^{3} = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
18.957 |
|
|
\[
{}y^{\prime \prime }+y y^{\prime }-y^{3}+a y = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
4.429 |
|
|
\[
{}y^{\prime \prime }+\left (y+3 a \right ) y^{\prime }-y^{3}+a y^{2}+2 a^{2} y = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
4.059 |
|
|
\[
{}y^{\prime \prime }+\left (y+3 f \left (x \right )\right ) y^{\prime }-y^{3}+y^{2} f \left (x \right )+y \left (f^{\prime }\left (x \right )+2 f \left (x \right )^{2}\right ) = 0
\] |
[NONE] |
✗ |
0.214 |
|
|
\[
{}y^{\prime \prime }+y y^{\prime }-y^{3}-\left (\frac {f^{\prime }\left (x \right )}{f \left (x \right )}+f \left (x \right )\right ) \left (3 y^{\prime }+y^{2}\right )+\left (a f \left (x \right )^{2}+3 f^{\prime }\left (x \right )+\frac {3 {f^{\prime }\left (x \right )}^{2}}{f \left (x \right )^{2}}-\frac {f^{\prime \prime }\left (x \right )}{f \left (x \right )}\right ) y+b f \left (x \right )^{3} = 0
\] |
[NONE] |
✗ |
0.578 |
|
|
\[
{}y^{\prime \prime }+\left (y-\frac {3 f^{\prime }\left (x \right )}{2 f \left (x \right )}\right ) y^{\prime }-y^{3}-\frac {f^{\prime }\left (x \right ) y^{2}}{2 f \left (x \right )}+\frac {\left (f \left (x \right )+\frac {{f^{\prime }\left (x \right )}^{2}}{f \left (x \right )^{2}}-f^{\prime \prime }\left (x \right )\right ) y}{2 f \left (x \right )} = 0
\] |
[NONE] |
✗ |
0.503 |
|
|
\[
{}y^{\prime \prime }+2 y y^{\prime }+f \left (x \right ) y^{\prime }+f^{\prime }\left (x \right ) y = 0
\] |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
2.292 |
|
|
\[
{}y^{\prime \prime }+2 y y^{\prime }+f \left (x \right ) \left (y^{\prime }+y^{2}\right )-g \left (x \right ) = 0
\] |
[[_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.206 |
|
|
\[
{}y^{\prime \prime }+3 y y^{\prime }+y^{3}+f \left (x \right ) y-g \left (x \right ) = 0
\] |
[NONE] |
✗ |
0.203 |
|
|
\[
{}y^{\prime \prime }+\left (3 y+f \left (x \right )\right ) y^{\prime }+y^{3}+y^{2} f \left (x \right ) = 0
\] |
[[_2nd_order, _with_potential_symmetries]] |
✗ |
0.193 |
|
|
\[
{}y^{\prime \prime }-3 y y^{\prime }-3 a y^{2}-4 a^{2} y-b = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
3.011 |
|
|
\[
{}y^{\prime \prime }-\left (3 y+f \left (x \right )\right ) y^{\prime }+y^{3}+y^{2} f \left (x \right ) = 0
\] |
[[_2nd_order, _with_potential_symmetries]] |
✗ |
0.197 |
|
|
\[
{}y^{\prime \prime }-2 a y y^{\prime } = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.674 |
|
|
\[
{}y^{\prime \prime }+a y y^{\prime }+b y^{3} = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
43.683 |
|
|
\[
{}y^{\prime \prime }+f \left (x , y\right ) y^{\prime }+g \left (x , y\right ) = 0
\] |
[NONE] |
✗ |
0.145 |
|
|
\[
{}y^{\prime \prime }+a {y^{\prime }}^{2}+b y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
0.989 |
|
|
\[
{}y^{\prime \prime }+a y^{\prime } {| y^{\prime }|}+b y^{\prime }+c y = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
1.701 |
|
|
\[
{}y^{\prime \prime }+a {y^{\prime }}^{2}+b y^{\prime }+c y = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
1.683 |
|
|
\[
{}y^{\prime \prime }+a {y^{\prime }}^{2}+b \sin \left (y\right ) = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
4.235 |
|
|
\[
{}y^{\prime \prime }+a y^{\prime } {| y^{\prime }|}+b \sin \left (y\right ) = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
3.080 |
|
|
\[
{}y^{\prime \prime }+a y {y^{\prime }}^{2}+b y = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
1.174 |
|
|
\[
{}y^{\prime \prime }+f \left (y\right ) {y^{\prime }}^{2}+g \left (x \right ) y^{\prime } = 0
\] |
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.441 |
|
|
\[
{}y^{\prime \prime }-\frac {D\left (f \right )\left (y\right ) {y^{\prime }}^{3}}{f \left (y\right )}+g \left (x \right ) y^{\prime }+h \left (x \right ) f \left (y\right ) = 0
\] |
[NONE] |
✗ |
0.250 |
|
|
\[
{}y^{\prime \prime }+\phi \left (y\right ) {y^{\prime }}^{2}+f \left (x \right ) y^{\prime }+g \left (x \right ) \Phi \left (y\right ) = 0
\] |
[NONE] |
✗ |
0.244 |
|
|
\[
{}y^{\prime \prime }+f \left (y\right ) {y^{\prime }}^{2}+g \left (y\right ) y^{\prime }+h \left (y\right ) = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
1.836 |
|
|
\[
{}y^{\prime \prime }+\left ({y^{\prime }}^{2}+1\right ) \left (f \left (x , y\right ) y^{\prime }+g \left (x , y\right )\right ) = 0
\] |
[NONE] |
✗ |
0.163 |
|
|
\[
{}y^{\prime \prime }+a y \left ({y^{\prime }}^{2}+1\right )^{2} = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
2.694 |
|
|
\[
{}y^{\prime \prime }-a \left (-y+y^{\prime } x \right )^{v} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.208 |
|
|
\[
{}y^{\prime \prime }-k \,x^{a} y^{b} {y^{\prime }}^{r} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.171 |
|
|
\[
{}y^{\prime \prime }+\left (y^{\prime }-\frac {y}{x}\right )^{a} f \left (x , y\right ) = 0
\] |
[NONE] |
✗ |
0.259 |
|
|
\[
{}y^{\prime \prime } = a \sqrt {{y^{\prime }}^{2}+1}
\] |
[[_2nd_order, _missing_x]] |
✓ |
5.923 |
|
|
\[
{}y^{\prime \prime } = a \sqrt {{y^{\prime }}^{2}+1}+b
\] |
[[_2nd_order, _missing_x]] |
✓ |
2.708 |
|
|
\[
{}y^{\prime \prime } = a \sqrt {{y^{\prime }}^{2}+b y^{2}}
\] |
[[_2nd_order, _missing_x]] |
✓ |
2.127 |
|
|
\[
{}y^{\prime \prime } = a \left ({y^{\prime }}^{2}+1\right )^{{3}/{2}}
\] |
[[_2nd_order, _missing_x]] |
✓ |
3.229 |
|
|
\[
{}y^{\prime \prime }-2 a x \left ({y^{\prime }}^{2}+1\right )^{{3}/{2}} = 0
\] |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
2.512 |
|
|
\[
{}y^{\prime \prime }-a y \left ({y^{\prime }}^{2}+1\right )^{{3}/{2}} = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
7.556 |
|
|
\[
{}y^{\prime \prime } = 2 a \left (c +b x +y\right ) \left ({y^{\prime }}^{2}+1\right )^{{3}/{2}}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
0.276 |
|
|
\[
{}y^{\prime \prime }+y^{3} y^{\prime }-y y^{\prime } \sqrt {y^{4}+4 y^{\prime }} = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
3.648 |
|
|
\[
{}y^{\prime \prime }-f \left (y^{\prime }, a x +b y\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.145 |
|
|
\[
{}y^{\prime \prime }-y f \left (x , \frac {y^{\prime }}{y}\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.146 |
|
|
\[
{}y^{\prime \prime }-x^{n -2} f \left (y x^{-n}, y^{\prime } x^{-n +1}\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.223 |
|
|
\[
{}8 y^{\prime \prime }+9 {y^{\prime }}^{4} = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
2.463 |
|
|
\[
{}a y^{\prime \prime }+h \left (y^{\prime }\right )+c y = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
2.342 |
|
|
\[
{}x y^{\prime \prime }+2 y^{\prime }-x y^{n} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.142 |
|
|
\[
{}x y^{\prime \prime }+2 y^{\prime }+a \,x^{v} y^{n} = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
0.155 |
|
|
\[
{}x y^{\prime \prime }+2 y^{\prime }+x \,{\mathrm e}^{y} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.146 |
|
|
\[
{}x y^{\prime \prime }+a y^{\prime }+b x \,{\mathrm e}^{y} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.149 |
|
|
\[
{}x y^{\prime \prime }+a y^{\prime }+b \,x^{5-2 a} {\mathrm e}^{y} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.183 |
|
|
\[
{}x y^{\prime \prime }+\left (-1+y\right ) y^{\prime } = 0
\] |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.535 |
|
|
\[
{}x y^{\prime \prime }-x^{2} {y^{\prime }}^{2}+2 y^{\prime }+y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.146 |
|
|
\[
{}x y^{\prime \prime }+a \left (-y+y^{\prime } x \right )^{2}-b = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.149 |
|
|
\[
{}2 x y^{\prime \prime }+{y^{\prime }}^{3}+y^{\prime } = 0
\] |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
0.584 |
|
|
\[
{}x^{2} y^{\prime \prime } = a \left (y^{n}-y\right )
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.149 |
|
|
\[
{}x^{2} y^{\prime \prime }+a \left ({\mathrm e}^{y}-1\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.157 |
|
|
\[
{}x^{2} y^{\prime \prime }-\left (2 a +b -1\right ) x y^{\prime }+\left (c^{2} b^{2} x^{2 b}+a \left (a +b \right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.555 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a +1\right ) x y^{\prime }-x^{k} f \left (x^{k} y, y^{\prime } x +k y\right ) = 0
\] |
[NONE] |
✗ |
0.211 |
|
|
\[
{}x^{2} y^{\prime \prime }+a \left (-y+y^{\prime } x \right )^{2}-b \,x^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.148 |
|
|
\[
{}x^{2} y^{\prime \prime }+a y {y^{\prime }}^{2}+b x = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.130 |
|
|
\[
{}x^{2} y^{\prime \prime }-\sqrt {a \,x^{2} {y^{\prime }}^{2}+b y^{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.270 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0
\] |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
0.719 |
|
|
\[
{}4 x^{2} y^{\prime \prime }-x^{4} {y^{\prime }}^{2}+4 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.133 |
|
|
\[
{}9 x^{2} y^{\prime \prime }+a y^{3}+2 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.174 |
|
|
\[
{}x^{3} \left (y^{\prime \prime }+y y^{\prime }-y^{3}\right )+12 x y+24 = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.161 |
|
|
\[
{}x^{3} y^{\prime \prime }-a \left (-y+y^{\prime } x \right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.160 |
|
|
\[
{}2 x^{3} y^{\prime \prime }+x^{2} \left (9+2 x y\right ) y^{\prime }+b +x y \left (a +3 x y-2 x^{2} y^{2}\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.167 |
|
|
\[
{}2 \left (-x^{k}+4 x^{3}\right ) \left (y^{\prime \prime }+y y^{\prime }-y^{3}\right )-\left (k \,x^{k -1}-12 x^{2}\right ) \left (3 y^{\prime }+y^{2}\right )+y a x +b = 0
\] |
[NONE] |
✗ |
0.283 |
|
|
\[
{}x^{4} y^{\prime \prime }+a^{2} y^{n} = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
0.141 |
|
|
\[
{}x^{4} y^{\prime \prime }-x \left (x^{2}+2 y\right ) y^{\prime }+4 y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.145 |
|
|
\[
{}x^{4} y^{\prime \prime }-x^{2} \left (x +y^{\prime }\right ) y^{\prime }+4 y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.151 |
|
|
\[
{}x^{4} y^{\prime \prime }+\left (-y+y^{\prime } x \right )^{3} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.140 |
|
|
\[
{}y^{\prime \prime } \sqrt {x}-y^{{3}/{2}} = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
0.161 |
|
|
\[
{}\left (a \,x^{2}+b x +c \right )^{{3}/{2}} y^{\prime \prime }-F \left (\frac {y}{\sqrt {a \,x^{2}+b x +c}}\right ) = 0
\] |
[NONE] |
✗ |
8.091 |
|
|
\[
{}x^{\frac {n}{n +1}} y^{\prime \prime }-y^{\frac {2 n +1}{n +1}} = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
0.200 |
|
|
\[
{}f \left (x \right )^{2} y^{\prime \prime }+f \left (x \right ) f^{\prime }\left (x \right ) y^{\prime }-h \left (y, f \left (x \right ) y^{\prime }\right ) = 0
\] |
[NONE] |
✗ |
0.227 |
|
|
\[
{}y^{\prime \prime } y-a = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
0.540 |
|
|
\[
{}y^{\prime \prime } y-a x = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
0.120 |
|
|
\[
{}y^{\prime \prime } y-a \,x^{2} = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
0.124 |
|
|
\[
{}y^{\prime \prime } y+{y^{\prime }}^{2}-a = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
1.632 |
|
|
\[
{}y^{\prime \prime } y+y^{2}-a x -b = 0
\] |
[NONE] |
✗ |
0.136 |
|
|
\[
{}y^{\prime \prime } y+{y^{\prime }}^{2}-y^{\prime } = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.613 |
|
|
\[
{}y^{\prime \prime } y-{y^{\prime }}^{2}+1 = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
2.060 |
|
|
\[
{}y^{\prime \prime } y-{y^{\prime }}^{2}-1 = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
2.809 |
|