|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}y^{\prime } = \frac {x \left (a -1\right ) \left (a +1\right )}{y+F \left (\frac {y^{2}}{2}-\frac {a^{2} x^{2}}{2}+\frac {x^{2}}{2}\right ) a^{2}-F \left (\frac {y^{2}}{2}-\frac {a^{2} x^{2}}{2}+\frac {x^{2}}{2}\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
2.493 |
|
|
\[
{}y^{\prime } = \frac {y}{x \left (-1+F \left (x y\right ) y\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
232.320 |
|
|
\[
{}y^{\prime } = -\frac {-x^{2}+2 x^{3} y-F \left (\left (x y-1\right ) x \right )}{x^{4}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
1.508 |
|
|
\[
{}y^{\prime } = \frac {F \left (\frac {\left (y+3\right ) {\mathrm e}^{\frac {3 x^{2}}{2}}}{3 y}\right ) x y^{2} {\mathrm e}^{3 x^{2}} {\mathrm e}^{-\frac {9 x^{2}}{2}}}{9}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
3.214 |
|
|
\[
{}y^{\prime } = \frac {\left (y+1\right ) \left (\left (y-\ln \left (y+1\right )-\ln \left (x \right )\right ) x +1\right )}{y x}
\] |
[‘y=_G(x,y’)‘] |
✗ |
3.967 |
|
|
\[
{}y^{\prime } = \frac {6 y}{8 y^{4}+9 y^{3}+12 y^{2}+6 y-F \left (-\frac {y^{4}}{3}-\frac {y^{3}}{2}-y^{2}-y+x \right )}
\] |
[‘x=_G(y,y’)‘] |
✗ |
2.777 |
|
|
\[
{}y^{\prime } = \frac {y^{2}+2 x y+x^{2}+{\mathrm e}^{2 F \left (-\left (x -y\right ) \left (x +y\right )\right )}}{y^{2}+2 x y+x^{2}-{\mathrm e}^{2 F \left (-\left (x -y\right ) \left (x +y\right )\right )}}
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
2.514 |
|
|
\[
{}y^{\prime } = \frac {1}{y+\sqrt {x}}
\] |
[[_homogeneous, ‘class G‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
2.537 |
|
|
\[
{}y^{\prime } = \frac {1}{y+2+\sqrt {3 x +1}}
\] |
[[_1st_order, _with_linear_symmetries], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
5.191 |
|
|
\[
{}y^{\prime } = \frac {x^{2}}{y+x^{{3}/{2}}}
\] |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
5.468 |
|
|
\[
{}y^{\prime } = \frac {x^{{5}/{3}}}{y+x^{{4}/{3}}}
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
4.710 |
|
|
\[
{}y^{\prime } = \frac {i x^{2} \left (i-2 \sqrt {-x^{3}+6 y}\right )}{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
36.171 |
|
|
\[
{}y^{\prime } = \frac {x}{y+\sqrt {x^{2}+1}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✗ |
4.415 |
|
|
\[
{}y^{\prime } = \frac {\left (-1+y \ln \left (x \right )\right )^{2}}{x}
\] |
[_Riccati] |
✓ |
2.446 |
|
|
\[
{}y^{\prime } = \frac {x \left (-2+3 \sqrt {x^{2}+3 y}\right )}{3}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
3.028 |
|
|
\[
{}y^{\prime } = \frac {\left (2 y \ln \left (x \right )-1\right )^{2}}{x}
\] |
[_Riccati] |
✓ |
2.769 |
|
|
\[
{}y^{\prime } = \frac {{\mathrm e}^{b x}}{y \,{\mathrm e}^{-b x}+1}
\] |
[[_1st_order, _with_linear_symmetries], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
3.391 |
|
|
\[
{}y^{\prime } = \frac {x^{2} \left (1+2 \sqrt {x^{3}-6 y}\right )}{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
3.608 |
|
|
\[
{}y^{\prime } = \frac {{\mathrm e}^{x}}{y \,{\mathrm e}^{-x}+1}
\] |
[[_1st_order, _with_linear_symmetries], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
3.342 |
|
|
\[
{}y^{\prime } = \frac {{\mathrm e}^{\frac {2 x}{3}}}{y \,{\mathrm e}^{-\frac {2 x}{3}}+1}
\] |
[[_1st_order, _with_linear_symmetries], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
3.067 |
|
|
\[
{}y^{\prime } = \frac {1+2 x^{5} \sqrt {4 x^{2} y+1}}{2 x^{3}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
36.740 |
|
|
\[
{}y^{\prime } = \frac {x \left (x +2 \sqrt {x^{3}-6 y}\right )}{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
3.711 |
|
|
\[
{}y^{\prime } = \left (-\ln \left (y\right )+x^{2}\right ) y
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
1.483 |
|
|
\[
{}y^{\prime } = \frac {{\mathrm e}^{-x^{2}} x}{y \,{\mathrm e}^{x^{2}}+1}
\] |
[[_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
2.764 |
|
|
\[
{}y^{\prime } = -\left (-\ln \left (\ln \left (y\right )\right )+\ln \left (x \right )\right ) y
\] |
[‘x=_G(y,y’)‘] |
✗ |
2.968 |
|
|
\[
{}y^{\prime } = \left (-\ln \left (\ln \left (y\right )\right )+\ln \left (x \right )\right )^{2} y
\] |
[‘y=_G(x,y’)‘] |
✗ |
3.476 |
|
|
\[
{}y^{\prime } = \frac {y}{\ln \left (\ln \left (y\right )\right )-\ln \left (x \right )+1}
\] |
[‘y=_G(x,y’)‘] |
✗ |
3.609 |
|
|
\[
{}y^{\prime } = \frac {1+2 \sqrt {4 x^{2} y+1}\, x^{4}}{2 x^{3}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
36.699 |
|
|
\[
{}y^{\prime } = \frac {\left (-y^{2}+4 a x \right )^{2}}{y}
\] |
[_rational] |
✗ |
2.860 |
|
|
\[
{}y^{\prime } = \frac {x \left (-2+3 x \sqrt {x^{2}+3 y}\right )}{3}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
3.030 |
|
|
\[
{}y^{\prime } = -\frac {x^{2} \left (a x -2 \sqrt {a \left (a \,x^{4}+8 y\right )}\right )}{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
5.619 |
|
|
\[
{}y^{\prime } = \left (-\ln \left (y\right )+x \right ) y
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
1.056 |
|
|
\[
{}y^{\prime } = \frac {x^{3}+x^{2}+2 \sqrt {x^{3}-6 y}}{2 x +2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
3.084 |
|
|
\[
{}y^{\prime } = \frac {\left (y^{2} a +b \,x^{2}\right )^{2} x}{a^{{5}/{2}} y}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
3.073 |
|
|
\[
{}y^{\prime } = -\frac {x^{3} \left (\sqrt {a}\, x +\sqrt {a}-2 \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2 \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
5.883 |
|
|
\[
{}y^{\prime } = -\frac {x}{4}+\frac {1}{4}+x \sqrt {x^{2}-2 x +1+8 y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
3.411 |
|
|
\[
{}y^{\prime } = -\frac {x}{2}-\frac {a}{2}+x \sqrt {x^{2}+2 a x +a^{2}+4 y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
3.212 |
|
|
\[
{}y^{\prime } = \frac {\left (\ln \left (y\right )+x^{2}\right ) y}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
1.536 |
|
|
\[
{}y^{\prime } = \frac {2 a +x \sqrt {-y^{2}+4 a x}}{y}
\] |
[‘y=_G(x,y’)‘] |
✗ |
37.254 |
|
|
\[
{}y^{\prime } = -\frac {x}{2}+1+x \sqrt {x^{2}-4 x +4 y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
3.374 |
|
|
\[
{}y^{\prime } = -\frac {2 x^{2}+2 x -3 \sqrt {x^{2}+3 y}}{3 \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
2.592 |
|
|
\[
{}y^{\prime } = \frac {y^{3} {\mathrm e}^{-\frac {4 x}{3}}}{y \,{\mathrm e}^{-\frac {2 x}{3}}+1}
\] |
[[_1st_order, _with_linear_symmetries], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
2.530 |
|
|
\[
{}y^{\prime } = \frac {\left (\ln \left (y\right )+x^{3}\right ) y}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
1.488 |
|
|
\[
{}y^{\prime } = -\frac {x}{4}+\frac {1}{4}+x^{2} \sqrt {x^{2}-2 x +1+8 y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
3.389 |
|
|
\[
{}y^{\prime } = -\frac {x^{2}-1-4 \sqrt {x^{2}-2 x +1+8 y}}{4 \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
2.432 |
|
|
\[
{}y^{\prime } = -\frac {a x}{2}-\frac {b}{2}+x \sqrt {a^{2} x^{2}+2 a b x +b^{2}+4 a y-4 c}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
3.262 |
|
|
\[
{}y^{\prime } = -\frac {x}{2}-\frac {a}{2}+x^{2} \sqrt {x^{2}+2 a x +a^{2}+4 y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
3.162 |
|
|
\[
{}y^{\prime } = -\frac {a x}{2}-\frac {b}{2}+x^{2} \sqrt {a^{2} x^{2}+2 a b x +b^{2}+4 a y-4 c}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
3.328 |
|
|
\[
{}y^{\prime } = \frac {x}{2}+\frac {1}{2}+x^{2} \sqrt {x^{2}+2 x +1-4 y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
3.184 |
|
|
\[
{}y^{\prime } = \frac {2 a +x^{2} \sqrt {-y^{2}+4 a x}}{y}
\] |
[‘y=_G(x,y’)‘] |
✗ |
35.528 |
|
|
\[
{}y^{\prime } = -\frac {x}{2}+1+x^{2} \sqrt {x^{2}-4 x +4 y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
3.434 |
|
|
\[
{}y^{\prime } = -\frac {\left (\sqrt {a}\, x^{4}+\sqrt {a}\, x^{3}-2 \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2 \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
37.449 |
|
|
\[
{}y^{\prime } = \left (-\ln \left (y\right )+1+x^{2}+x^{3}\right ) y
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
2.493 |
|
|
\[
{}y^{\prime } = \frac {y^{3} {\mathrm e}^{-2 b x}}{y \,{\mathrm e}^{-b x}+1}
\] |
[[_1st_order, _with_linear_symmetries], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
1.746 |
|
|
\[
{}y^{\prime } = \frac {y^{3} {\mathrm e}^{-2 x}}{y \,{\mathrm e}^{-x}+1}
\] |
[[_1st_order, _with_linear_symmetries], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
2.549 |
|
|
\[
{}y^{\prime } = \frac {\left (-2 y^{{3}/{2}}+3 \,{\mathrm e}^{x}\right )^{2} {\mathrm e}^{x}}{4 \sqrt {y}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
4.018 |
|
|
\[
{}y^{\prime } = \frac {i x \left (i-2 \sqrt {-x^{2}+4 \ln \left (a \right )+4 \ln \left (y\right )}\right ) y}{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
5.522 |
|
|
\[
{}y^{\prime } = \frac {\left (x y^{2}+1\right )^{2}}{y x^{4}}
\] |
[_rational] |
✗ |
2.663 |
|
|
\[
{}y^{\prime } = \frac {x^{2} \left (3 x +\sqrt {-9 x^{4}+4 y^{3}}\right )}{y^{2}}
\] |
[‘y=_G(x,y’)‘] |
✗ |
45.205 |
|
|
\[
{}y^{\prime } = \frac {-\sin \left (2 y\right )+\cos \left (2 y\right ) x^{2}+x^{2}}{2 x}
\] |
[‘y=_G(x,y’)‘] |
✓ |
2.314 |
|
|
\[
{}y^{\prime } = -\frac {x^{2}-x -2-2 \sqrt {x^{2}-4 x +4 y}}{2 \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
2.561 |
|
|
\[
{}y^{\prime } = \frac {y+x^{3} a \,{\mathrm e}^{x}+a \,x^{4}+a \,x^{3}-x y^{2} {\mathrm e}^{x}-x^{2} y^{2}-x y^{2}}{x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
2.553 |
|
|
\[
{}y^{\prime } = \frac {x +1+2 x^{6} \sqrt {4 x^{2} y+1}}{2 x^{3} \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
6.581 |
|
|
\[
{}y^{\prime } = \frac {y+x^{3} a \ln \left (x +1\right )+a \,x^{4}+a \,x^{3}-x y^{2} \ln \left (x +1\right )-x^{2} y^{2}-x y^{2}}{x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
2.826 |
|
|
\[
{}y^{\prime } = \frac {x^{2} \left (x +1+2 x \sqrt {x^{3}-6 y}\right )}{2 x +2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
5.085 |
|
|
\[
{}y^{\prime } = \frac {y+x^{3} \ln \left (x \right )+x^{4}+x^{3}+7 x y^{2} \ln \left (x \right )+7 x^{2} y^{2}+7 x y^{2}}{x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
5.296 |
|
|
\[
{}y^{\prime } = \frac {x^{2}+2 x +1+2 \sqrt {x^{2}+2 x +1-4 y}}{2 x +2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
2.437 |
|
|
\[
{}y^{\prime } = \frac {y+x^{3} b \ln \left (\frac {1}{x}\right )+x^{4} b +b \,x^{3}+x a y^{2} \ln \left (\frac {1}{x}\right )+x^{2} a y^{2}+a x y^{2}}{x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
5.579 |
|
|
\[
{}y^{\prime } = \frac {2 a}{x \left (-x y+2 a x y^{2}-8 a^{2}\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
2.258 |
|
|
\[
{}y^{\prime } = \frac {y \left (-1+\ln \left (x \left (x +1\right )\right ) y x^{4}-\ln \left (x \left (x +1\right )\right ) x^{3}\right )}{x}
\] |
[_Bernoulli] |
✓ |
5.179 |
|
|
\[
{}y^{\prime } = \frac {y+\sqrt {y^{2}+x^{2}}\, x^{2}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
8.983 |
|
|
\[
{}y^{\prime } = \frac {y+\ln \left (\left (x -1\right ) \left (x +1\right )\right ) x^{3}+7 \ln \left (\left (x -1\right ) \left (x +1\right )\right ) x y^{2}}{x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
2.898 |
|
|
\[
{}y^{\prime } = \frac {y^{3} x \,{\mathrm e}^{2 x^{2}}}{y \,{\mathrm e}^{x^{2}}+1}
\] |
[[_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
3.013 |
|
|
\[
{}y^{\prime } = \frac {y-\ln \left (\frac {x +1}{x -1}\right ) x^{3}+\ln \left (\frac {x +1}{x -1}\right ) x y^{2}}{x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
3.150 |
|
|
\[
{}y^{\prime } = \frac {y+{\mathrm e}^{\frac {x +1}{x -1}} x^{3}+{\mathrm e}^{\frac {x +1}{x -1}} x y^{2}}{x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
3.253 |
|
|
\[
{}y^{\prime } = \frac {x y-y-{\mathrm e}^{x +1} x^{3}+{\mathrm e}^{x +1} x y^{2}}{\left (x -1\right ) x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
2.520 |
|
|
\[
{}y^{\prime } = \frac {-x^{2}+1+4 x^{3} \sqrt {x^{2}-2 x +1+8 y}}{4 x +4}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
5.997 |
|
|
\[
{}y^{\prime } = \frac {-\sin \left (2 y\right )+\cos \left (2 y\right ) x^{3}+x^{3}}{2 x}
\] |
[‘y=_G(x,y’)‘] |
✓ |
2.504 |
|
|
\[
{}y^{\prime } = \frac {y+x^{3} \sqrt {y^{2}+x^{2}}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
3.072 |
|
|
\[
{}y^{\prime } = \left (1+y^{2} {\mathrm e}^{-2 b x}+y^{3} {\mathrm e}^{-3 b x}\right ) {\mathrm e}^{b x}
\] |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
1.616 |
|
|
\[
{}y^{\prime } = \frac {x +1+2 \sqrt {4 x^{2} y+1}\, x^{3}}{2 x^{3} \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
5.288 |
|
|
\[
{}y^{\prime } = \frac {y \ln \left (x -1\right )+x^{4}+x^{3}+x^{2} y^{2}+x y^{2}}{\ln \left (x -1\right ) x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
2.724 |
|
|
\[
{}y^{\prime } = \frac {y \ln \left (x -1\right )+{\mathrm e}^{x +1} x^{3}+7 \,{\mathrm e}^{x +1} x y^{2}}{\ln \left (x -1\right ) x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
3.714 |
|
|
\[
{}y^{\prime } = \left (1+y^{2} {\mathrm e}^{-\frac {4 x}{3}}+y^{3} {\mathrm e}^{-2 x}\right ) {\mathrm e}^{\frac {2 x}{3}}
\] |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
2.192 |
|
|
\[
{}y^{\prime } = \left (1+y^{2} {\mathrm e}^{-2 x}+y^{3} {\mathrm e}^{-3 x}\right ) {\mathrm e}^{x}
\] |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
1.661 |
|
|
\[
{}y^{\prime } = \frac {x \left (-2 x -2+3 x^{2} \sqrt {x^{2}+3 y}\right )}{3+3 x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
5.233 |
|
|
\[
{}y^{\prime } = \frac {1}{x \left (x y^{2}+1+x \right ) y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
1.795 |
|
|
\[
{}y^{\prime } = \frac {2 x \,{\mathrm e}^{x}-2 x -\ln \left (x \right )-1+x^{4} \ln \left (x \right )+x^{4}-2 y x^{2} \ln \left (x \right )-2 x^{2} y+y^{2} \ln \left (x \right )+y^{2}}{{\mathrm e}^{x}-1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✗ |
17.802 |
|
|
\[
{}y^{\prime } = \frac {-y \,{\mathrm e}^{x}+x y-x^{3} \ln \left (x \right )-x^{3}-x y^{2} \ln \left (x \right )-x y^{2}}{\left (-{\mathrm e}^{x}+x \right ) x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
3.354 |
|
|
\[
{}y^{\prime } = \frac {y \left (1-x +y x^{2} \ln \left (x \right )+x^{3} y-x \ln \left (x \right )-x^{2}\right )}{\left (x -1\right ) x}
\] |
[_Bernoulli] |
✓ |
3.230 |
|
|
\[
{}y^{\prime } = \frac {y \ln \left (x \right ) x -y+2 x^{5} b +2 x^{3} a y^{2}}{\left (x \ln \left (x \right )-1\right ) x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
3.109 |
|
|
\[
{}y^{\prime } = \frac {\left (\ln \left (y\right )+x +x^{3}+x^{4}\right ) y}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
1.949 |
|
|
\[
{}y^{\prime } = -\frac {\left (-\ln \left (y-1\right )+\ln \left (y+1\right )+2 \ln \left (x \right )\right ) x \left (y+1\right )^{2}}{8}
\] |
[‘y=_G(x,y’)‘] |
✗ |
20.075 |
|
|
\[
{}y^{\prime } = \frac {\left (-\ln \left (y-1\right )+\ln \left (y+1\right )+2 \ln \left (x \right )\right )^{2} x \left (y+1\right )^{2}}{16}
\] |
[‘y=_G(x,y’)‘] |
✗ |
44.592 |
|
|
\[
{}y^{\prime } = \frac {\left (-y^{2}+4 a x \right )^{3}}{\left (-y^{2}+4 a x -1\right ) y}
\] |
[_rational] |
✗ |
2.840 |
|
|
\[
{}y^{\prime } = \frac {2 a x +2 a +x^{3} \sqrt {-y^{2}+4 a x}}{\left (x +1\right ) y}
\] |
[‘y=_G(x,y’)‘] |
✗ |
43.694 |
|
|
\[
{}y^{\prime } = \frac {-\ln \left (x \right )+{\mathrm e}^{\frac {1}{x}}+4 x^{2} y+2 x +2 x y^{2}+2 x^{3}}{\ln \left (x \right )-{\mathrm e}^{\frac {1}{x}}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
211.440 |
|
|
\[
{}y^{\prime } = -\frac {\left (\ln \left (y\right ) x +\ln \left (y\right )-1\right ) y}{x +1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
2.881 |
|
|
\[
{}y^{\prime } = \frac {x^{2}+2 x +1+2 x^{3} \sqrt {x^{2}+2 x +1-4 y}}{2 x +2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
6.314 |
|
|
\[
{}y^{\prime } = \frac {-b y a +b^{2}+a b +b^{2} x -b a \sqrt {x}-a^{2}}{a \left (-a y+b +a +b x -a \sqrt {x}\right )}
\] |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.366 |
|