|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}y^{\prime } = \frac {2 x \ln \left (\frac {1}{x -1}\right )-\coth \left (\frac {x +1}{x -1}\right )+\coth \left (\frac {x +1}{x -1}\right ) y^{2}-2 \coth \left (\frac {x +1}{x -1}\right ) x^{2} y+\coth \left (\frac {x +1}{x -1}\right ) x^{4}}{\ln \left (\frac {1}{x -1}\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✗ |
326.544 |
|
|
\[
{}y^{\prime } = \frac {2 x^{2} \cosh \left (\frac {1}{x -1}\right )-2 x \cosh \left (\frac {1}{x -1}\right )-1+y^{2}-2 x^{2} y+x^{4}-x +x y^{2}-2 x^{3} y+x^{5}}{\left (x -1\right ) \cosh \left (\frac {1}{x -1}\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✗ |
148.700 |
|
|
\[
{}y^{\prime } = \frac {y \left (-\cosh \left (\frac {1}{x +1}\right ) x +\cosh \left (\frac {1}{x +1}\right )-x +x^{2} y-x^{2}+x^{3} y\right )}{x \left (x -1\right ) \cosh \left (\frac {1}{x +1}\right )}
\] |
[_Bernoulli] |
✓ |
8.960 |
|
|
\[
{}y^{\prime } = -\frac {y \left (x y+1\right )}{x \left (x y+1-y\right )}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
2.014 |
|
|
\[
{}y^{\prime } = \frac {y}{x \left (-1+y+x^{2} y^{3}+y^{4} x^{3}\right )}
\] |
[_rational] |
✗ |
7.382 |
|
|
\[
{}y^{\prime } = \frac {x^{3}+3 a \,x^{2}+3 a^{2} x +a^{3}+x y^{2}+a y^{2}+y^{3}}{\left (x +a \right )^{3}}
\] |
[[_homogeneous, ‘class C‘], _rational, _Abel] |
✓ |
74.157 |
|
|
\[
{}y^{\prime } = \frac {y^{3} x \,{\mathrm e}^{3 x^{2}} {\mathrm e}^{-\frac {9 x^{2}}{2}}}{9 \,{\mathrm e}^{\frac {3 x^{2}}{2}}+3 \,{\mathrm e}^{\frac {3 x^{2}}{2}} y+9 y}
\] |
[[_Abel, ‘2nd type‘, ‘class C‘]] |
✗ |
3.980 |
|
|
\[
{}y^{\prime } = \frac {y \left (-1-\cosh \left (\frac {x +1}{x -1}\right ) x +\cosh \left (\frac {x +1}{x -1}\right ) x^{2} y-\cosh \left (\frac {x +1}{x -1}\right ) x^{2}+\cosh \left (\frac {x +1}{x -1}\right ) x^{3} y\right )}{x}
\] |
[_Bernoulli] |
✓ |
13.647 |
|
|
\[
{}y^{\prime } = \frac {\left (x +y+1\right ) y}{\left (2 y^{3}+y+x \right ) \left (x +1\right )}
\] |
[_rational] |
✗ |
2.502 |
|
|
\[
{}y^{\prime } = \frac {y \left (-1-x \,{\mathrm e}^{\frac {x +1}{x -1}}+x^{2} {\mathrm e}^{\frac {x +1}{x -1}} y-x^{2} {\mathrm e}^{\frac {x +1}{x -1}}+x^{3} {\mathrm e}^{\frac {x +1}{x -1}} y\right )}{x}
\] |
[_Bernoulli] |
✓ |
5.164 |
|
|
\[
{}y^{\prime } = \frac {-b^{3}+6 b^{2} x -12 b \,x^{2}+8 x^{3}-4 b y^{2}+8 x y^{2}+8 y^{3}}{\left (2 x -b \right )^{3}}
\] |
[[_homogeneous, ‘class C‘], _rational, _Abel] |
✓ |
75.132 |
|
|
\[
{}y^{\prime } = \frac {\left (y \,{\mathrm e}^{-\frac {x^{2}}{4}} x +2+2 y^{2} {\mathrm e}^{-\frac {x^{2}}{2}}+2 y^{3} {\mathrm e}^{-\frac {3 x^{2}}{4}}\right ) {\mathrm e}^{\frac {x^{2}}{4}}}{2}
\] |
[_Abel] |
✓ |
4.008 |
|
|
\[
{}y^{\prime } = -\frac {-\frac {1}{x}-\textit {\_F1} \left (y+\frac {1}{x}\right )}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
2.301 |
|
|
\[
{}y^{\prime } = \frac {\textit {\_F1} \left (y^{2}-2 \ln \left (x \right )\right )}{\sqrt {y^{2}}\, x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
2.946 |
|
|
\[
{}y^{\prime } = \frac {-\sin \left (2 y\right ) x -\sin \left (2 y\right )+\cos \left (2 y\right ) x^{4}+x^{4}}{2 x \left (x +1\right )}
\] |
[‘y=_G(x,y’)‘] |
✗ |
18.680 |
|
|
\[
{}y^{\prime } = \frac {x y+y+x^{4} \sqrt {x^{2}+y^{2}}}{x \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
7.134 |
|
|
\[
{}y^{\prime } = \frac {-\sin \left (2 y\right ) x -\sin \left (2 y\right )+x \cos \left (2 y\right )+x}{2 x \left (x +1\right )}
\] |
[‘y=_G(x,y’)‘] |
✗ |
10.594 |
|
|
\[
{}y^{\prime } = -\frac {1}{-x -\textit {\_F1} \left (y-\ln \left (x \right )\right ) y \,{\mathrm e}^{y}}
\] |
[NONE] |
✗ |
3.105 |
|
|
\[
{}y^{\prime } = \frac {\left (1+2 y\right ) \left (1+y\right )}{x \left (-2 y-2+x +2 x y\right )}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
2.191 |
|
|
\[
{}y^{\prime } = \frac {-125+300 x -240 x^{2}+64 x^{3}-80 y^{2}+64 x y^{2}+64 y^{3}}{\left (4 x -5\right )^{3}}
\] |
[[_homogeneous, ‘class C‘], _rational, _Abel] |
✓ |
81.249 |
|
|
\[
{}y^{\prime } = \frac {x +y+y^{2}-2 y \ln \left (x \right ) x +x^{2} \ln \left (x \right )^{2}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
1.926 |
|
|
\[
{}y^{\prime } = \frac {x^{3} {\mathrm e}^{y}+x^{4}+{\mathrm e}^{y} y-{\mathrm e}^{y} \ln \left ({\mathrm e}^{y}+x \right )+x y-\ln \left ({\mathrm e}^{y}+x \right ) x +x}{x^{2}}
\] |
[‘y=_G(x,y’)‘] |
✗ |
38.453 |
|
|
\[
{}y^{\prime } = \frac {x^{2}}{2}+\sqrt {x^{3}-6 y}+x^{2} \sqrt {x^{3}-6 y}+x^{3} \sqrt {x^{3}-6 y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
45.353 |
|
|
\[
{}y^{\prime } = \frac {\left (-\sqrt {a}\, x^{3}+2 \sqrt {a \,x^{4}+8 y}+2 x^{2} \sqrt {a \,x^{4}+8 y}+2 x^{3} \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
38.826 |
|
|
\[
{}y^{\prime } = \frac {y \left (-3 x^{3} y-3+y^{2} x^{7}\right )}{x \left (x^{3} y+1\right )}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
2.454 |
|
|
\[
{}y^{\prime } = \frac {\left (3+y\right )^{3} {\mathrm e}^{\frac {9 x^{2}}{2}} x \,{\mathrm e}^{\frac {3 x^{2}}{2}} {\mathrm e}^{-3 x^{2}}}{243 \,{\mathrm e}^{\frac {3 x^{2}}{2}}+81 \,{\mathrm e}^{\frac {3 x^{2}}{2}} y+243 y}
\] |
[[_Abel, ‘2nd type‘, ‘class C‘]] |
✗ |
4.541 |
|
|
\[
{}y^{\prime } = \frac {\left (x -y\right )^{3} \left (x +y\right )^{3} x}{\left (-y^{2}+x^{2}-1\right ) y}
\] |
[_rational] |
✗ |
2.484 |
|
|
\[
{}y^{\prime } = \frac {-2 \cos \left (y\right )+x^{3} \cos \left (2 y\right ) \ln \left (x \right )+x^{3} \ln \left (x \right )}{2 \sin \left (y\right ) \ln \left (x \right ) x}
\] |
[‘y=_G(x,y’)‘] |
✗ |
55.089 |
|
|
\[
{}y^{\prime } = \frac {y}{x \left (-1+x y+x y^{3}+x y^{4}\right )}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
2.733 |
|
|
\[
{}y^{\prime } = -\frac {2 x}{3}+\sqrt {x^{2}+3 y}+x^{2} \sqrt {x^{2}+3 y}+x^{3} \sqrt {x^{2}+3 y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
36.572 |
|
|
\[
{}y^{\prime } = \frac {-2 \cos \left (y\right )+x^{2} \cos \left (2 y\right ) \ln \left (x \right )+x^{2} \ln \left (x \right )}{2 \sin \left (y\right ) \ln \left (x \right ) x}
\] |
[‘y=_G(x,y’)‘] |
✗ |
54.163 |
|
|
\[
{}y^{\prime } = \frac {y \left (x y+1\right )}{x \left (-x y-1+y^{4} x^{3}\right )}
\] |
[_rational] |
✓ |
2.332 |
|
|
\[
{}y^{\prime } = \frac {\left (4 \,{\mathrm e}^{-x^{2}}-4 x^{2} {\mathrm e}^{-x^{2}}+4 y^{2}-4 x^{2} {\mathrm e}^{-x^{2}} y+x^{4} {\mathrm e}^{-2 x^{2}}\right ) x}{4}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
2.750 |
|
|
\[
{}y^{\prime } = \frac {y \left (x +y\right )}{x \left (x +y+y^{3}+y^{4}\right )}
\] |
[_rational] |
✓ |
1.497 |
|
|
\[
{}y^{\prime } = \frac {y \left (x^{3}+x^{2} y+y^{2}\right )}{x^{2} \left (x -1\right ) \left (x +y\right )}
\] |
[[_homogeneous, ‘class D‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
3.784 |
|
|
\[
{}y^{\prime } = \frac {\left (\left (x^{2}+1\right )^{{3}/{2}} x^{2}+\left (x^{2}+1\right )^{{3}/{2}}+y^{2} \left (x^{2}+1\right )^{{3}/{2}}+x^{2} y^{3}+y^{3}\right ) x}{\left (x^{2}+1\right )^{3}}
\] |
[_Abel] |
✗ |
17.180 |
|
|
\[
{}y^{\prime } = \frac {\left (3 x y^{2}+x +3 y^{2}\right ) y}{\left (6 y^{2}+x \right ) x \left (x +1\right )}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
2.698 |
|
|
\[
{}y^{\prime } = -\frac {-y+x^{3} \sqrt {x^{2}+y^{2}}-x^{2} \sqrt {x^{2}+y^{2}}\, y}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
6.810 |
|
|
\[
{}y^{\prime } = \frac {\left (1+2 y\right ) \left (1+y\right )}{x \left (-2 y-2+x y^{3}+2 x y^{4}\right )}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
2.863 |
|
|
\[
{}y^{\prime } = \frac {1+2 \sqrt {4 x^{2} y+1}\, x^{3}+2 x^{5} \sqrt {4 x^{2} y+1}+2 x^{6} \sqrt {4 x^{2} y+1}}{2 x^{3}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
41.716 |
|
|
\[
{}y^{\prime } = \frac {y \left (x -y\right )}{x \left (x -y-y^{3}-y^{4}\right )}
\] |
[_rational] |
✓ |
1.519 |
|
|
\[
{}y^{\prime } = \frac {2 a +\sqrt {-y^{2}+4 a x}+x^{2} \sqrt {-y^{2}+4 a x}+x^{3} \sqrt {-y^{2}+4 a x}}{y}
\] |
[‘y=_G(x,y’)‘] |
✗ |
52.494 |
|
|
\[
{}y^{\prime } = \frac {\left (x +y+1\right ) y}{\left (y^{4}+y^{3}+y^{2}+x \right ) \left (x +1\right )}
\] |
[_rational] |
✗ |
2.372 |
|
|
\[
{}y^{\prime } = -\frac {-y+x^{4} \sqrt {x^{2}+y^{2}}-x^{3} \sqrt {x^{2}+y^{2}}\, y}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
7.263 |
|
|
\[
{}y^{\prime } = \frac {\left (x^{4}+3 x y^{2}+3 y^{2}\right ) y}{\left (6 y^{2}+x \right ) x \left (x +1\right )}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
3.179 |
|
|
\[
{}y^{\prime } = -\frac {1}{-\left (y^{3}\right )^{{2}/{3}} x -\textit {\_F1} \left (y^{3}-3 \ln \left (x \right )\right ) \left (y^{3}\right )^{{1}/{3}} x}
\] |
[NONE] |
✗ |
2.308 |
|
|
\[
{}y^{\prime } = \frac {y \left (x -y\right ) \left (1+y\right )}{x \left (x y+x -y\right )}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
1.656 |
|
|
\[
{}y^{\prime } = -\frac {1}{-\ln \left (x \right ) \left (y^{3}\right )^{{2}/{3}}-\textit {\_F1} \left (y^{3}+3 \,\operatorname {Ei}_{1}\left (-\ln \left (x \right )\right )\right ) \ln \left (x \right ) \left (y^{3}\right )^{{1}/{3}}}
\] |
[NONE] |
✗ |
69.822 |
|
|
\[
{}y^{\prime } = \frac {30 x^{3}+25 \sqrt {x}+25 y^{2}-20 x^{3} y-100 y \sqrt {x}+4 x^{6}+40 x^{{7}/{2}}+100 x}{25 x}
\] |
[_rational, _Riccati] |
✓ |
16.383 |
|
|
\[
{}y^{\prime } = \frac {\left ({\mathrm e}^{-\frac {y}{x}} y+{\mathrm e}^{-\frac {y}{x}} x +x^{2}\right ) {\mathrm e}^{\frac {y}{x}}}{x}
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
2.002 |
|
|
\[
{}y^{\prime } = \frac {\left ({\mathrm e}^{-\frac {y}{x}} y+{\mathrm e}^{-\frac {y}{x}} x +x^{3}\right ) {\mathrm e}^{\frac {y}{x}}}{x}
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
1.850 |
|
|
\[
{}y^{\prime } = \frac {b \,x^{3}+c^{2} \sqrt {a}-2 c b \,x^{2} \sqrt {a}+2 c y^{2} a^{{3}/{2}}+b^{2} x^{4} \sqrt {a}-2 y^{2} a^{{3}/{2}} b \,x^{2}+a^{{5}/{2}} y^{4}}{a \,x^{2} y}
\] |
[_rational] |
✗ |
5.559 |
|
|
\[
{}y^{\prime } = \frac {y+x^{2} \ln \left (x \right )^{3}+2 x^{2} \ln \left (x \right )^{2} y+x^{2} \ln \left (x \right ) y^{2}}{x \ln \left (x \right )}
\] |
[_Riccati] |
✓ |
6.635 |
|
|
\[
{}y^{\prime } = \frac {y+x^{3} \ln \left (x \right )^{3}+2 x^{3} \ln \left (x \right )^{2} y+x^{3} \ln \left (x \right ) y^{2}}{x \ln \left (x \right )}
\] |
[_Riccati] |
✓ |
3.074 |
|
|
\[
{}y^{\prime } = \frac {y \left (x +y\right ) \left (1+y\right )}{x \left (x y+x +y\right )}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
1.616 |
|
|
\[
{}y^{\prime } = \frac {3 x^{3}+\sqrt {-9 x^{4}+4 y^{3}}+x^{2} \sqrt {-9 x^{4}+4 y^{3}}+x^{3} \sqrt {-9 x^{4}+4 y^{3}}}{y^{2}}
\] |
[NONE] |
✗ |
38.362 |
|
|
\[
{}y^{\prime } = \frac {1}{-x +\left (\frac {1}{y}+1\right ) x +\textit {\_F1} \left (\left (\frac {1}{y}+1\right ) x \right ) x^{2}-\textit {\_F1} \left (\left (\frac {1}{y}+1\right ) x \right ) x^{2} \left (\frac {1}{y}+1\right )}
\] |
[‘y=_G(x,y’)‘] |
✗ |
2.219 |
|
|
\[
{}y^{\prime } = \frac {x}{2}+\frac {1}{2}+\sqrt {x^{2}+2 x +1-4 y}+x^{2} \sqrt {x^{2}+2 x +1-4 y}+x^{3} \sqrt {x^{2}+2 x +1-4 y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
36.978 |
|
|
\[
{}y^{\prime } = \frac {\cosh \left (x \right )}{\sinh \left (x \right )}+\textit {\_F1} \left (y-\ln \left (\sinh \left (x \right )\right )\right )
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
4.740 |
|
|
\[
{}y^{\prime } = -\frac {x}{2}+1+\sqrt {x^{2}-4 x +4 y}+x^{2} \sqrt {x^{2}-4 x +4 y}+x^{3} \sqrt {x^{2}-4 x +4 y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
36.986 |
|
|
\[
{}y^{\prime } = \frac {1}{\sin \left (x \right )}+\textit {\_F1} \left (y-\ln \left (\sin \left (x \right )\right )+\ln \left (\cos \left (x \right )+1\right )\right )
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
7.696 |
|
|
\[
{}y^{\prime } = \frac {b^{3}+y^{2} b^{3}+2 y b^{2} a x +x^{2} b \,a^{2}+y^{3} b^{3}+3 y^{2} b^{2} a x +3 y b \,a^{2} x^{2}+a^{3} x^{3}}{b^{3}}
\] |
[[_homogeneous, ‘class C‘], _Abel] |
✓ |
10.010 |
|
|
\[
{}y^{\prime } = \frac {\alpha ^{3}+y^{2} \alpha ^{3}+2 y \alpha ^{2} \beta x +\alpha \,\beta ^{2} x^{2}+y^{3} \alpha ^{3}+3 y^{2} \alpha ^{2} \beta x +3 y \alpha \,\beta ^{2} x^{2}+\beta ^{3} x^{3}}{\alpha ^{3}}
\] |
[[_homogeneous, ‘class C‘], _Abel] |
✓ |
10.155 |
|
|
\[
{}y^{\prime } = \frac {14 x y+12+2 x +x^{3} y^{3}+6 x^{2} y^{2}}{x^{2} \left (x y+2+x \right )}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
2.112 |
|
|
\[
{}y^{\prime } = \frac {y \left (\ln \left (x \right )+\ln \left (y\right )-1+x^{2} \ln \left (x \right )^{2}+2 x^{2} \ln \left (y\right ) \ln \left (x \right )+x^{2} \ln \left (y\right )^{2}\right )}{x}
\] |
[NONE] |
✗ |
3.743 |
|
|
\[
{}y^{\prime } = \frac {y \left (\ln \left (y\right )-1+\ln \left (x \right )+x^{3} \ln \left (x \right )^{2}+2 x^{3} \ln \left (y\right ) \ln \left (x \right )+x^{3} \ln \left (y\right )^{2}\right )}{x}
\] |
[NONE] |
✗ |
3.820 |
|
|
\[
{}y^{\prime } = -\frac {\left (-\frac {1}{x}-\textit {\_F1} \left (y^{2}-2 x \right )\right ) x}{\sqrt {y^{2}}}
\] |
[NONE] |
✗ |
2.347 |
|
|
\[
{}y^{\prime } = -\frac {x}{4}+\frac {1}{4}+\sqrt {x^{2}-2 x +1+8 y}+x^{2} \sqrt {x^{2}-2 x +1+8 y}+x^{3} \sqrt {x^{2}-2 x +1+8 y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
37.215 |
|
|
\[
{}y^{\prime } = \frac {a^{3}+y^{2} a^{3}+2 y a^{2} b x +a \,b^{2} x^{2}+y^{3} a^{3}+3 y^{2} a^{2} b x +3 y a \,b^{2} x^{2}+b^{3} x^{3}}{a^{3}}
\] |
[[_homogeneous, ‘class C‘], _Abel] |
✓ |
10.066 |
|
|
\[
{}y^{\prime } = -\frac {-x -\textit {\_F1} \left (y^{2}-2 x \right )}{\sqrt {y^{2}}\, x}
\] |
[NONE] |
✗ |
2.520 |
|
|
\[
{}y^{\prime } = \frac {-\sin \left (2 y\right )+x \cos \left (2 y\right )+\cos \left (2 y\right ) x^{3}+\cos \left (2 y\right ) x^{4}+x +x^{3}+x^{4}}{2 x}
\] |
[‘y=_G(x,y’)‘] |
✓ |
3.786 |
|
|
\[
{}y^{\prime } = -\frac {\left (-\frac {y \,{\mathrm e}^{\frac {1}{x}}}{x}-\textit {\_F1} \left (y \,{\mathrm e}^{\frac {1}{x}}\right )\right ) {\mathrm e}^{-\frac {1}{x}}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
2.817 |
|
|
\[
{}y^{\prime } = \frac {y+x \sqrt {x^{2}+y^{2}}+x^{3} \sqrt {x^{2}+y^{2}}+x^{4} \sqrt {x^{2}+y^{2}}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
7.842 |
|
|
\[
{}y^{\prime } = \frac {y \left ({\mathrm e}^{-\frac {x^{2}}{2}} x y+{\mathrm e}^{-\frac {x^{2}}{4}} x +2 y^{2} {\mathrm e}^{-\frac {3 x^{2}}{4}}\right ) {\mathrm e}^{\frac {x^{2}}{4}}}{2 y \,{\mathrm e}^{-\frac {x^{2}}{4}}+2}
\] |
[[_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
3.738 |
|
|
\[
{}y^{\prime } = \left (\frac {\ln \left (-1+y\right ) y}{\left (1-y\right ) \ln \left (x \right ) x}-\frac {\ln \left (-1+y\right )}{\left (1-y\right ) \ln \left (x \right ) x}-f \left (x \right )\right ) \left (1-y\right )
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
3.496 |
|
|
\[
{}y^{\prime } = -\frac {x}{2}-\frac {a}{2}+\sqrt {x^{2}+2 a x +a^{2}+4 y}+x^{2} \sqrt {x^{2}+2 a x +a^{2}+4 y}+x^{3} \sqrt {x^{2}+2 a x +a^{2}+4 y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
38.645 |
|
|
\[
{}y^{\prime } = -\frac {2 x}{3}+1+y^{2}+\frac {2 x^{2} y}{3}+\frac {x^{4}}{9}+y^{3}+x^{2} y^{2}+\frac {y x^{4}}{3}+\frac {x^{6}}{27}
\] |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
1.275 |
|
|
\[
{}y^{\prime } = 2 x +1+y^{2}-2 x^{2} y+x^{4}+y^{3}-3 x^{2} y^{2}+3 y x^{4}-x^{6}
\] |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
1.188 |
|
|
\[
{}y^{\prime } = \frac {-x +1-2 y+3 x^{2}-2 x^{2} y+2 x^{4}+x^{3}-2 x^{3} y+2 x^{5}}{x^{2}-y}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
2.342 |
|
|
\[
{}y^{\prime } = \frac {\left ({\mathrm e}^{-\frac {y}{x}} y+{\mathrm e}^{-\frac {y}{x}} x +x +x^{3}+x^{4}\right ) {\mathrm e}^{\frac {y}{x}}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
4.109 |
|
|
\[
{}y^{\prime } = \frac {2 x y^{2}+4 y \ln \left (2 x +1\right ) x +2 \ln \left (2 x +1\right )^{2} x +y^{2}-2+\ln \left (2 x +1\right )^{2}+2 y \ln \left (2 x +1\right )}{2 x +1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
2.319 |
|
|
\[
{}y^{\prime } = \frac {-30 x^{3} y+12 x^{6}+70 x^{{7}/{2}}-30 x^{3}-25 y \sqrt {x}+50 x -25 \sqrt {x}-25}{5 \left (-5 y+2 x^{3}+10 \sqrt {x}-5\right ) x}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
2.358 |
|
|
\[
{}y^{\prime } = \frac {1+2 y}{x \left (-2+x +x y^{2}+3 x y^{3}+2 x y+2 x y^{4}\right )}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
2.787 |
|
|
\[
{}y^{\prime } = \frac {\left (-256 a \,x^{2}+512+512 y^{2}+128 y a \,x^{4}+8 a^{2} x^{8}+512 y^{3}+192 x^{4} a y^{2}+24 y a^{2} x^{8}+a^{3} x^{12}\right ) x}{512}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
1.917 |
|
|
\[
{}y^{\prime } = -\frac {-x y-y+x^{5} \sqrt {x^{2}+y^{2}}-x^{4} \sqrt {x^{2}+y^{2}}\, y}{x \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
35.549 |
|
|
\[
{}y^{\prime } = -\frac {y^{2} \left (x^{2} y-2 x -2 x y+y\right )}{2 \left (-2+x y-2 y\right ) x}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
2.882 |
|
|
\[
{}y^{\prime } = \frac {-2 x y+2 x^{3}-2 x -y^{3}+3 x^{2} y^{2}-3 y x^{4}+x^{6}}{-y+x^{2}-1}
\] |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
1.849 |
|
|
\[
{}y^{\prime } = \frac {1+y^{4}-8 a x y^{2}+16 a^{2} x^{2}+y^{6}-12 y^{4} a x +48 y^{2} a^{2} x^{2}-64 a^{3} x^{3}}{y}
\] |
[_rational] |
✗ |
3.102 |
|
|
\[
{}y^{\prime } = -\frac {-x y-y+\sqrt {x^{2}+y^{2}}\, x^{2}-x \sqrt {x^{2}+y^{2}}\, y}{x \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
9.454 |
|
|
\[
{}y^{\prime } = -\frac {2 a}{-y-2 a -2 a y^{4}+16 a^{2} x y^{2}-32 a^{3} x^{2}-2 a y^{6}+24 y^{4} a^{2} x -96 y^{2} a^{3} x^{2}+128 a^{4} x^{3}}
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
1.935 |
|
|
\[
{}y^{\prime } = \frac {-18 x y-6 x^{3}-18 x +27 y^{3}+27 x^{2} y^{2}+9 y x^{4}+x^{6}}{27 y+9 x^{2}+27}
\] |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
1.885 |
|
|
\[
{}y^{\prime } = -\frac {\left (-108 x^{{3}/{2}}-216-216 y^{2}+72 x^{3} y-6 x^{6}-216 y^{3}+108 x^{3} y^{2}-18 y x^{6}+x^{9}\right ) \sqrt {x}}{216}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
7.347 |
|
|
\[
{}y^{\prime } = \frac {\left (a^{3}+y^{4} a^{3}+2 y^{2} a^{2} b \,x^{2}+a \,x^{4} b^{2}+y^{6} a^{3}+3 y^{4} a^{2} b \,x^{2}+3 y^{2} a \,b^{2} x^{4}+b^{3} x^{6}\right ) x}{a^{{7}/{2}} y}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
4.510 |
|
|
\[
{}y^{\prime } = -\frac {\left (-1-y^{4}+2 x^{2} y^{2}-x^{4}-y^{6}+3 x^{2} y^{4}-3 y^{2} x^{4}+x^{6}\right ) x}{y}
\] |
[_rational] |
✗ |
2.546 |
|
|
\[
{}y^{\prime } = -\frac {i \left (32 i x +64+64 y^{4}+32 x^{2} y^{2}+4 x^{4}+64 y^{6}+48 x^{2} y^{4}+12 y^{2} x^{4}+x^{6}\right )}{128 y}
\] |
[_rational] |
✗ |
3.153 |
|
|
\[
{}y^{\prime } = \frac {2 x^{2}-4 x^{3} y+1+y^{2} x^{4}+x^{6} y^{3}-3 y^{2} x^{5}+3 y x^{4}-x^{3}}{x^{4}}
\] |
[_rational, _Abel] |
✓ |
2.173 |
|
|
\[
{}y^{\prime } = \frac {y a^{2} x +a +a^{2} x +y^{3} a^{3} x^{3}+3 y^{2} a^{2} x^{2}+3 y a x +1}{a^{2} x^{2} \left (y a x +1+a x \right )}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
2.124 |
|
|
\[
{}y^{\prime } = \frac {6 x^{2} y-2 x +1-5 x^{3} y^{2}-2 x y+y^{3} x^{4}}{x^{2} \left (x^{2} y-x +1\right )}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
2.706 |
|
|
\[
{}y^{\prime } = -\frac {\left (-8-8 y^{3}+24 y^{{3}/{2}} {\mathrm e}^{x}-18 \,{\mathrm e}^{2 x}-8 y^{{9}/{2}}+36 y^{3} {\mathrm e}^{x}-54 y^{{3}/{2}} {\mathrm e}^{2 x}+27 \,{\mathrm e}^{3 x}\right ) {\mathrm e}^{x}}{8 \sqrt {y}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
46.330 |
|
|
\[
{}y^{\prime } = \frac {x}{-y+1+y^{4}+2 x^{2} y^{2}+x^{4}+y^{6}+3 x^{2} y^{4}+3 y^{2} x^{4}+x^{6}}
\] |
[_rational] |
✗ |
2.494 |
|