|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}y^{\prime } = -\frac {-y^{3}-y+4 y^{2} \ln \left (x \right )-4 \ln \left (x \right )^{2} y^{3}-1+6 y \ln \left (x \right )-12 \ln \left (x \right )^{2} y^{2}+8 \ln \left (x \right )^{3} y^{3}}{y x}
\] |
[[_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
3.599 |
|
|
\[
{}y^{\prime } = \frac {y \left (\ln \left (y\right ) x +\ln \left (y\right )-x -1+x \ln \left (x \right )+\ln \left (x \right )+x^{4} \ln \left (x \right )^{2}+2 x^{4} \ln \left (y\right ) \ln \left (x \right )+x^{4} \ln \left (y\right )^{2}\right )}{x \left (x +1\right )}
\] |
[NONE] |
✗ |
5.338 |
|
|
\[
{}y^{\prime } = \frac {y \left (x \ln \left (x \right )+\ln \left (x \right )+\ln \left (y\right ) x +\ln \left (y\right )-x -1+x \ln \left (x \right )^{2}+2 x \ln \left (y\right ) \ln \left (x \right )+x \ln \left (y\right )^{2}\right )}{x \left (x +1\right )}
\] |
[NONE] |
✗ |
4.645 |
|
|
\[
{}y^{\prime } = \frac {2 y^{8}}{y^{5}+2 y^{6}+2 y^{2}+16 x y^{4}+32 y^{6} x^{2}+2+24 x y^{2}+96 x^{2} y^{4}+128 x^{3} y^{6}}
\] |
[_rational] |
✓ |
2.172 |
|
|
\[
{}y^{\prime } = \frac {y^{{3}/{2}} \left (x -y+\sqrt {y}\right )}{y^{{3}/{2}} x -y^{{5}/{2}}+y^{2}+x^{3}-3 x^{2} y+3 x y^{2}-y^{3}}
\] |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
5.573 |
|
|
\[
{}y^{\prime } = \frac {2 y^{6} \left (1+4 x y^{2}+y^{2}\right )}{y^{3}+4 y^{5} x +y^{5}+2+24 x y^{2}+96 x^{2} y^{4}+128 x^{3} y^{6}}
\] |
[_rational] |
✗ |
3.157 |
|
|
\[
{}y^{\prime } = -\left (-\frac {\ln \left (y\right )}{x}+\frac {\ln \left (y\right )}{x \ln \left (x \right )}-\textit {\_F1} \left (x \right )\right ) y
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
37.381 |
|
|
\[
{}y^{\prime } = \frac {y^{2}}{y^{2}+y^{{3}/{2}}+\sqrt {y}\, x^{2}-2 y^{{3}/{2}} x +y^{{5}/{2}}+x^{3}-3 x^{2} y+3 x y^{2}-y^{3}}
\] |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
2.150 |
|
|
\[
{}y^{\prime } = \frac {y^{2}+2 x y+x^{2}+{\mathrm e}^{-2 \left (x -y\right ) \left (x +y\right )}}{y^{2}+2 x y+x^{2}-{\mathrm e}^{-2 \left (x -y\right ) \left (x +y\right )}}
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
2.179 |
|
|
\[
{}y^{\prime } = -\frac {\left (-\frac {\ln \left (y\right )^{2}}{2 x}-\textit {\_F1} \left (x \right )\right ) y}{\ln \left (y\right )}
\] |
[NONE] |
✓ |
7.806 |
|
|
\[
{}y^{\prime } = \frac {y^{2}+2 x y+x^{2}+{\mathrm e}^{2 \left (x -y\right )^{2} \left (x +y\right )^{2}}}{y^{2}+2 x y+x^{2}-{\mathrm e}^{2 \left (x -y\right )^{2} \left (x +y\right )^{2}}}
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
2.418 |
|
|
\[
{}y^{\prime } = \frac {-8 x^{2} y^{3}+16 x y^{2}+16 x y^{3}-8+12 x y-6 y^{2} x^{2}+x^{3} y^{3}}{16 \left (-2+x y-2 y\right ) x}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
3.038 |
|
|
\[
{}y^{\prime } = -\frac {\left (-8 \,{\mathrm e}^{-x^{2}}+8 x^{2} {\mathrm e}^{-x^{2}}-8-8 y^{2}+8 x^{2} {\mathrm e}^{-x^{2}} y-2 x^{4} {\mathrm e}^{-2 x^{2}}-8 y^{3}+12 x^{2} {\mathrm e}^{-x^{2}} y^{2}-6 y x^{4} {\mathrm e}^{-2 x^{2}}+x^{6} {\mathrm e}^{-3 x^{2}}\right ) x}{8}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
6.871 |
|
|
\[
{}y^{\prime } = \frac {\left ({\mathrm e}^{-\frac {y}{x}} y x +{\mathrm e}^{-\frac {y}{x}} y+{\mathrm e}^{-\frac {y}{x}} x^{2}+{\mathrm e}^{-\frac {y}{x}} x +x \right ) {\mathrm e}^{\frac {y}{x}}}{x \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
6.660 |
|
|
\[
{}y^{\prime } = -\frac {16 x y^{3}-8 y^{3}-8 y+8 x y^{2}-2 x^{2} y^{3}-8+12 x y-6 y^{2} x^{2}+x^{3} y^{3}}{32 y x}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
2.897 |
|
|
\[
{}y^{\prime } = \frac {\left ({\mathrm e}^{-\frac {y}{x}} y x +{\mathrm e}^{-\frac {y}{x}} y+{\mathrm e}^{-\frac {y}{x}} x^{2}+{\mathrm e}^{-\frac {y}{x}} x +x^{4}\right ) {\mathrm e}^{\frac {y}{x}}}{x \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
3.625 |
|
|
\[
{}y^{\prime } = \frac {-3 x^{2} y-2 x^{3}-2 x -x y^{2}-y+x^{3} y^{3}+3 x^{4} y^{2}+3 x^{5} y+x^{6}}{x \left (x y+x^{2}+1\right )}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
3.763 |
|
|
\[
{}y^{\prime } = \frac {\left (27 y^{3}+27 \,{\mathrm e}^{3 x^{2}} y+18 \,{\mathrm e}^{3 x^{2}} y^{2}+3 y^{3} {\mathrm e}^{3 x^{2}}+27 \,{\mathrm e}^{\frac {9 x^{2}}{2}}+27 \,{\mathrm e}^{\frac {9 x^{2}}{2}} y+9 \,{\mathrm e}^{\frac {9 x^{2}}{2}} y^{2}+{\mathrm e}^{\frac {9 x^{2}}{2}} y^{3}\right ) {\mathrm e}^{3 x^{2}} x \,{\mathrm e}^{-\frac {9 x^{2}}{2}}}{243 y}
\] |
[[_Abel, ‘2nd type‘, ‘class C‘]] |
✗ |
213.555 |
|
|
\[
{}y^{\prime } = -\frac {-x^{2}-x y-x^{3}-x y^{2}+2 y x^{2} \ln \left (x \right )-x^{3} \ln \left (x \right )^{2}-y^{3}+3 x y^{2} \ln \left (x \right )-3 x^{2} \ln \left (x \right )^{2} y+x^{3} \ln \left (x \right )^{3}}{x^{2}}
\] |
[_Abel] |
✗ |
3.726 |
|
|
\[
{}y^{\prime } = \frac {x}{2}+1+y^{2}+\frac {x^{2} y}{4}-x y-\frac {x^{4}}{8}+\frac {x^{3}}{8}+\frac {x^{2}}{4}+y^{3}-\frac {3 y^{2} x^{2}}{4}-\frac {3 x y^{2}}{2}+\frac {3 y x^{4}}{16}+\frac {3 x^{3} y}{4}-\frac {x^{6}}{64}-\frac {3 x^{5}}{32}
\] |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
1.643 |
|
|
\[
{}y^{\prime } = -\frac {x}{2}+1+y^{2}+\frac {7 x^{2} y}{2}-2 x y+\frac {13 x^{4}}{16}-\frac {3 x^{3}}{2}+x^{2}+y^{3}+\frac {3 y^{2} x^{2}}{4}-3 x y^{2}+\frac {3 y x^{4}}{16}-\frac {3 x^{3} y}{2}+\frac {x^{6}}{64}-\frac {3 x^{5}}{16}
\] |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
2.137 |
|
|
\[
{}y^{\prime } = -\frac {x}{4}+1+y^{2}+\frac {7 x^{2} y}{16}-\frac {x y}{2}+\frac {5 x^{4}}{128}-\frac {5 x^{3}}{64}+\frac {x^{2}}{16}+y^{3}+\frac {3 y^{2} x^{2}}{8}-\frac {3 x y^{2}}{4}+\frac {3 y x^{4}}{64}-\frac {3 x^{3} y}{16}+\frac {x^{6}}{512}-\frac {3 x^{5}}{256}
\] |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
1.660 |
|
|
\[
{}y^{\prime } = \frac {-2 y-2 \ln \left (2 x +1\right )-2+2 x y^{3}+y^{3}+6 y^{2} \ln \left (2 x +1\right ) x +3 y^{2} \ln \left (2 x +1\right )+6 y \ln \left (2 x +1\right )^{2} x +3 y \ln \left (2 x +1\right )^{2}+2 \ln \left (2 x +1\right )^{3} x +\ln \left (2 x +1\right )^{3}}{\left (2 x +1\right ) \left (y+\ln \left (2 x +1\right )+1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✗ |
38.647 |
|
|
\[
{}y^{\prime } = \frac {-x^{2}+x +1+y^{2}+5 x^{2} y-2 x y+4 x^{4}-3 x^{3}+y^{3}+3 y^{2} x^{2}-3 x y^{2}+3 y x^{4}-6 x^{3} y+x^{6}-3 x^{5}}{x}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
2.511 |
|
|
\[
{}y^{\prime } = \frac {-32 x y+16 x^{3}+16 x^{2}-32 x -64 y^{3}+48 y^{2} x^{2}+96 x y^{2}-12 y x^{4}-48 x^{3} y-48 x^{2} y+x^{6}+6 x^{5}+12 x^{4}}{-64 y+16 x^{2}+32 x -64}
\] |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
2.333 |
|
|
\[
{}y^{\prime } = \frac {y \ln \left (x \right ) x +x^{2} \ln \left (x \right )-2 x y-x^{2}-y^{2}-y^{3}+3 x y^{2} \ln \left (x \right )-3 x^{2} \ln \left (x \right )^{2} y+x^{3} \ln \left (x \right )^{3}}{x \left (-y+x \ln \left (x \right )-x \right )}
\] |
[[_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
5.046 |
|
|
\[
{}y^{\prime } = \frac {-32 x y-72 x^{3}+32 x^{2}-32 x +64 y^{3}+48 y^{2} x^{2}-192 x y^{2}+12 y x^{4}-96 x^{3} y+192 x^{2} y+x^{6}-12 x^{5}+48 x^{4}}{64 y+16 x^{2}-64 x +64}
\] |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
1.982 |
|
|
\[
{}y^{\prime } = -\frac {y^{2}+2 x y+x^{2}+{\mathrm e}^{\frac {2 \left (x -y\right )^{3} \left (x +y\right )^{3}}{-y^{2}+x^{2}-1}}}{-y^{2}-2 x y-x^{2}+{\mathrm e}^{\frac {2 \left (x -y\right )^{3} \left (x +y\right )^{3}}{-y^{2}+x^{2}-1}}}
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
11.404 |
|
|
\[
{}y^{\prime } = \frac {-128 x y-24 x^{3}+32 x^{2}-128 x +512 y^{3}+192 y^{2} x^{2}-384 x y^{2}+24 y x^{4}-96 x^{3} y+96 x^{2} y+x^{6}-6 x^{5}+12 x^{4}}{512 y+64 x^{2}-128 x +512}
\] |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
2.220 |
|
|
\[
{}y^{\prime } = \frac {-32 a x y-8 a^{2} x^{3}-16 a \,x^{2} b -32 a x +64 y^{3}+48 x^{2} a y^{2}+96 y^{2} b x +12 y a^{2} x^{4}+48 a b y x^{3}+48 y b^{2} x^{2}+a^{3} x^{6}+6 a^{2} x^{5} b +12 a \,x^{4} b^{2}+8 b^{3} x^{3}}{64 y+16 x^{2} a +32 b x +64}
\] |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
2.231 |
|
|
\[
{}y^{\prime } = \frac {-32 x y-8 x^{3}-16 x^{2} a -32 x +64 y^{3}+48 y^{2} x^{2}+96 a x y^{2}+12 y x^{4}+48 y a \,x^{3}+48 a^{2} x^{2} y+x^{6}+6 x^{5} a +12 a^{2} x^{4}+8 a^{3} x^{3}}{64 y+16 x^{2}+32 a x +64}
\] |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
1.845 |
|
|
\[
{}y^{\prime } = \frac {\left (-8 \,{\mathrm e}^{-x^{2}} y+4 x^{2} {\mathrm e}^{-2 x^{2}}-8 \,{\mathrm e}^{-x^{2}}+8 x^{2} {\mathrm e}^{-x^{2}} y-4 x^{4} {\mathrm e}^{-2 x^{2}}+8 x^{2} {\mathrm e}^{-x^{2}}-8 y^{3}+12 x^{2} {\mathrm e}^{-x^{2}} y^{2}-6 y x^{4} {\mathrm e}^{-2 x^{2}}+x^{6} {\mathrm e}^{-3 x^{2}}\right ) x}{-8 y+4 x^{2} {\mathrm e}^{-x^{2}}-8}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✗ |
45.455 |
|
|
\[
{}y^{\prime } = \frac {2 x^{2} \cos \left (x \right )+2 \sin \left (x \right ) x^{3}-2 x \sin \left (x \right )+2 x +2 y^{2} x^{2}-4 y \sin \left (x \right ) x +4 y \cos \left (x \right ) x^{2}+4 x y+3-\cos \left (2 x \right )-2 \sin \left (2 x \right ) x -4 \sin \left (x \right )+x^{2} \cos \left (2 x \right )+x^{2}+4 \cos \left (x \right ) x}{2 x^{3}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
23.091 |
|
|
\[
{}y^{\prime } = -\frac {216 y}{-216 y^{4}-252 y^{3}-396 y^{2}-216 y+36 x^{2}-72 x y+60 y^{5}-36 x y^{3}-72 x y^{2}-24 x y^{4}+4 y^{8}+12 y^{7}+33 y^{6}}
\] |
[_rational] |
✓ |
3.375 |
|
|
\[
{}y^{\prime } = \frac {x^{2} y+x^{4}+2 x^{3}-3 x^{2}+x y+x +y^{3}+3 y^{2} x^{2}-3 x y^{2}+3 y x^{4}-6 x^{3} y+x^{6}-3 x^{5}}{x \left (y+x^{2}-x +1\right )}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
2.679 |
|
|
\[
{}y^{\prime } = -\frac {a x}{2}+1+y^{2}+\frac {a \,x^{2} y}{2}+b x y+\frac {a^{2} x^{4}}{16}+\frac {a \,x^{3} b}{4}+\frac {b^{2} x^{2}}{4}+y^{3}+\frac {3 x^{2} a y^{2}}{4}+\frac {3 y^{2} b x}{2}+\frac {3 y a^{2} x^{4}}{16}+\frac {3 a b y x^{3}}{4}+\frac {3 y b^{2} x^{2}}{4}+\frac {a^{3} x^{6}}{64}+\frac {3 a^{2} x^{5} b}{32}+\frac {3 a \,x^{4} b^{2}}{16}+\frac {b^{3} x^{3}}{8}
\] |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
1.514 |
|
|
\[
{}y^{\prime } = -\frac {x}{2}+1+y^{2}+\frac {x^{2} y}{2}+a x y+\frac {x^{4}}{16}+\frac {a \,x^{3}}{4}+\frac {a^{2} x^{2}}{4}+y^{3}+\frac {3 y^{2} x^{2}}{4}+\frac {3 a x y^{2}}{2}+\frac {3 y x^{4}}{16}+\frac {3 y a \,x^{3}}{4}+\frac {3 a^{2} x^{2} y}{4}+\frac {x^{6}}{64}+\frac {3 x^{5} a}{32}+\frac {3 a^{2} x^{4}}{16}+\frac {a^{3} x^{3}}{8}
\] |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
1.740 |
|
|
\[
{}y^{\prime } = -\frac {-y+\sqrt {y^{2}+x^{2}}\, x^{2}-x \sqrt {y^{2}+x^{2}}\, y+x^{4} \sqrt {y^{2}+x^{2}}-x^{3} \sqrt {y^{2}+x^{2}}\, y+x^{5} \sqrt {y^{2}+x^{2}}-x^{4} \sqrt {y^{2}+x^{2}}\, y}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
44.024 |
|
|
\[
{}y^{\prime } = \frac {y \left (\ln \left (x \right )+\ln \left (y\right )-1+x \ln \left (x \right )^{2}+2 x \ln \left (y\right ) \ln \left (x \right )+x \ln \left (y\right )^{2}+x^{3} \ln \left (x \right )^{2}+2 x^{3} \ln \left (y\right ) \ln \left (x \right )+x^{3} \ln \left (y\right )^{2}+x^{4} \ln \left (x \right )^{2}+2 x^{4} \ln \left (y\right ) \ln \left (x \right )+x^{4} \ln \left (y\right )^{2}\right )}{x}
\] |
[NONE] |
✗ |
14.672 |
|
|
\[
{}y^{\prime } = \frac {150 x^{3}+125 \sqrt {x}+125+125 y^{2}-100 x^{3} y-500 \sqrt {x}\, y+20 x^{6}+200 x^{{7}/{2}}+500 x +125 y^{3}-150 x^{3} y^{2}-750 y^{2} \sqrt {x}+60 y x^{6}+600 y x^{{7}/{2}}+1500 x y-8 x^{9}-120 x^{{13}/{2}}-600 x^{4}-1000 x^{{3}/{2}}}{125 x}
\] |
[_rational, _Abel] |
✓ |
5.088 |
|
|
\[
{}y^{\prime } = \frac {-150 x^{3} y+60 x^{6}+350 x^{{7}/{2}}-150 x^{3}-125 \sqrt {x}\, y+250 x -125 \sqrt {x}-125 y^{3}+150 x^{3} y^{2}+750 y^{2} \sqrt {x}-60 y x^{6}-600 y x^{{7}/{2}}-1500 x y+8 x^{9}+120 x^{{13}/{2}}+600 x^{4}+1000 x^{{3}/{2}}}{25 \left (-5 y+2 x^{3}+10 \sqrt {x}-5\right ) x}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✗ |
31.224 |
|
|
\[
{}y^{\prime } = \frac {y \left (-1-x^{\frac {2}{\ln \left (x \right )+1}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{\ln \left (x \right )+1}} x^{2}-x^{\frac {2}{\ln \left (x \right )+1}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{\ln \left (x \right )+1}} x^{2} \ln \left (x \right )+x^{\frac {2}{\ln \left (x \right )+1}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{\ln \left (x \right )+1}} x^{2} y+2 x^{\frac {2}{\ln \left (x \right )+1}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{\ln \left (x \right )+1}} x^{2} y \ln \left (x \right )+x^{\frac {2}{\ln \left (x \right )+1}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{\ln \left (x \right )+1}} x^{2} y \ln \left (x \right )^{2}\right )}{\left (\ln \left (x \right )+1\right ) x}
\] |
[_Bernoulli] |
✗ |
54.734 |
|
|
\[
{}y^{\prime } = \frac {y \left (-1-x^{3} x^{\frac {2}{\ln \left (x \right )+1}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{\ln \left (x \right )+1}}-x^{3} x^{\frac {2}{\ln \left (x \right )+1}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{\ln \left (x \right )+1}} \ln \left (x \right )+x^{3} x^{\frac {2}{\ln \left (x \right )+1}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{\ln \left (x \right )+1}} y+2 x^{3} x^{\frac {2}{\ln \left (x \right )+1}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{\ln \left (x \right )+1}} y \ln \left (x \right )+x^{3} x^{\frac {2}{\ln \left (x \right )+1}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{\ln \left (x \right )+1}} y \ln \left (x \right )^{2}\right )}{\left (\ln \left (x \right )+1\right ) x}
\] |
[_Bernoulli] |
✗ |
55.700 |
|
|
\[
{}y^{\prime } = \frac {2 x +4 y \ln \left (2 x +1\right ) x +6 y^{2} \ln \left (2 x +1\right ) x +6 y \ln \left (2 x +1\right )^{2} x +2 \ln \left (2 x +1\right )^{3} x +2 x y^{3}+2 \ln \left (2 x +1\right )^{2} x +2 x y^{2}-1+3 y^{2} \ln \left (2 x +1\right )+3 y \ln \left (2 x +1\right )^{2}+y^{2}+y^{3}+2 y \ln \left (2 x +1\right )+\ln \left (2 x +1\right )^{2}+\ln \left (2 x +1\right )^{3}}{2 x +1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✗ |
4.587 |
|
|
\[
{}y^{\prime } = \frac {-y \sin \left (\frac {y}{x}\right )+y \sin \left (\frac {3 y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right )+y \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )+2 \sin \left (\frac {y}{x}\right ) x^{3} \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )}{2 \cos \left (\frac {y}{x}\right ) \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x}
\] |
[[_homogeneous, ‘class D‘]] |
✓ |
72.394 |
|
|
\[
{}y^{\prime } = \frac {-y \sin \left (\frac {y}{x}\right )+y \sin \left (\frac {3 y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right )+y \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )+2 \sin \left (\frac {y}{x}\right ) x^{2} \sin \left (\frac {y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right )}{2 \cos \left (\frac {y}{x}\right ) \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x}
\] |
[[_homogeneous, ‘class D‘]] |
✓ |
10.759 |
|
|
\[
{}y^{\prime } = \frac {y^{2}+2 x y+x^{2}+{\mathrm e}^{2+2 y^{4}-4 y^{2} x^{2}+2 x^{4}+2 y^{6}-6 x^{2} y^{4}+6 x^{4} y^{2}-2 x^{6}}}{y^{2}+2 x y+x^{2}-{\mathrm e}^{2+2 y^{4}-4 y^{2} x^{2}+2 x^{4}+2 y^{6}-6 x^{2} y^{4}+6 x^{4} y^{2}-2 x^{6}}}
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
3.579 |
|
|
\[
{}y^{\prime } = \frac {4 x \left (a -1\right ) \left (a +1\right ) \left (-y^{2}+a^{2} x^{2}-x^{2}-2\right )}{-4 y^{3}+4 a^{2} x^{2} y-4 x^{2} y-8 y-a^{2} y^{6}+3 a^{4} y^{4} x^{2}-6 y^{4} a^{2} x^{2}-3 a^{6} y^{2} x^{4}+9 y^{2} a^{4} x^{4}-9 y^{2} a^{2} x^{4}+a^{8} x^{6}-4 a^{6} x^{6}+6 a^{4} x^{6}-4 a^{2} x^{6}+y^{6}+3 x^{2} y^{4}+3 x^{4} y^{2}+x^{6}}
\] |
[_rational] |
✗ |
5.856 |
|
|
\[
{}y^{\prime } = \frac {-4 \cos \left (x \right ) x +4 \sin \left (x \right ) x^{2}+4 x +4+4 y^{2}+8 y \cos \left (x \right ) x -8 x y+2 x^{2} \cos \left (2 x \right )+6 x^{2}-8 x^{2} \cos \left (x \right )+4 y^{3}+12 y^{2} \cos \left (x \right ) x -12 x y^{2}+6 y x^{2} \cos \left (2 x \right )+18 x^{2} y-24 y \cos \left (x \right ) x^{2}+x^{3} \cos \left (3 x \right )+15 x^{3} \cos \left (x \right )-6 x^{3} \cos \left (2 x \right )-10 x^{3}}{4 x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
20.331 |
|
|
\[
{}y^{\prime } = -\frac {8 x \left (a -1\right ) \left (a +1\right )}{8+x^{6}+2 x^{4}-8 a^{2}-2 a^{6} x^{4}+6 a^{4} x^{4}-6 a^{2} x^{4}+2 y^{4}+4 y^{2} x^{2}-8 y+3 x^{2} y^{4}+3 x^{4} y^{2}-2 a^{2} y^{4}+y^{6}+4 a^{4} y^{2} x^{2}+3 a^{4} y^{4} x^{2}-3 a^{6} y^{2} x^{4}+9 y^{2} a^{4} x^{4}-9 y^{2} a^{2} x^{4}-6 y^{4} a^{2} x^{2}-a^{2} y^{6}+a^{8} x^{6}-4 a^{6} x^{6}+6 a^{4} x^{6}-8 y^{2} a^{2} x^{2}-4 a^{2} x^{6}}
\] |
[_rational] |
✗ |
5.958 |
|
|
\[
{}y^{\prime } = \frac {-y \sin \left (\frac {y}{x}\right )+y \sin \left (\frac {3 y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right )+y \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )+2 \sin \left (\frac {y}{x}\right ) \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x +2 \sin \left (\frac {y}{x}\right ) x^{3} \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )+2 \sin \left (\frac {y}{x}\right ) x^{4} \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )}{2 \cos \left (\frac {y}{x}\right ) \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x}
\] |
[[_homogeneous, ‘class D‘]] |
✓ |
22.652 |
|
|
\[
{}y^{\prime } = -\frac {1296 y}{216+216 x^{3}-324 x^{2} y^{3}-432 x y+216 x y^{2}-1944 y^{4}-648 y^{2} x^{2}+1152 x y^{4}-1728 y^{3}-1296 y+216 x^{2}-2376 y^{2}-216 x^{2} y^{4}-648 x^{2} y+1080 x y^{3}-570 y^{8}-612 y^{5}+1080 y^{5} x -882 y^{6}-846 y^{7}-126 y^{10}-315 y^{9}-8 y^{12}-36 y^{11}+72 y^{8} x +216 y^{7} x +594 x y^{6}}
\] |
[_rational] |
✓ |
2.995 |
|
|
\[
{}y^{\prime } = -\frac {x \left (-513-432 x -456 x^{6}-576 x^{5}-756 x^{3}-648 x^{2} y^{3}-864 x^{4}-144 x^{7}-1296 y^{2} x^{2}+1008 x^{5} y-216 y^{3}-96 x^{8}-378 y-1134 x^{2}-540 y^{2}+64 x^{9}-594 x^{2} y+432 x^{3} y^{2}+720 x^{3} y-216 x^{6} y^{3}-216 x^{6} y^{2}-972 x^{4} y^{2}+432 y^{2} x^{7}-648 y^{3} x^{4}-288 y x^{8}+288 y x^{7}-216 y x^{4}+864 y^{2} x^{5}-288 y x^{6}\right )}{216 \left (x^{2}+1\right )^{4}}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
3.219 |
|
|
\[
{}y^{\prime } = \frac {-\sin \left (\frac {y}{x}\right ) y x -y \sin \left (\frac {y}{x}\right )+y \sin \left (\frac {3 y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right ) x +y \sin \left (\frac {3 y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right )+y \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x +y \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )+2 \sin \left (\frac {y}{x}\right ) x^{4} \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )}{2 \cos \left (\frac {y}{x}\right ) \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x \left (x +1\right )}
\] |
[[_homogeneous, ‘class D‘]] |
✓ |
44.636 |
|
|
\[
{}y^{\prime } = \frac {y \sin \left (\frac {3 y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right ) x +y \sin \left (\frac {3 y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right )+y \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x +y \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )-\sin \left (\frac {y}{x}\right ) y x -y \sin \left (\frac {y}{x}\right )+2 \sin \left (\frac {y}{x}\right ) \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x}{2 \cos \left (\frac {y}{x}\right ) \sin \left (\frac {y}{2 x}\right ) x \cos \left (\frac {y}{2 x}\right ) \left (x +1\right )}
\] |
[[_homogeneous, ‘class D‘]] |
✓ |
40.921 |
|
|
\[
{}y^{\prime } = -\frac {216 y \left (-2 y^{4}-3 y^{3}-6 y^{2}-6 y+6 x +6\right )}{216 x^{3}-324 x^{2} y^{3}-1296 x y-1944 x y^{2}+2808 y^{4}-648 y^{2} x^{2}-432 x y^{4}+1728 y^{3}-1296 y-1296 y^{2}-216 x^{2} y^{4}-648 x^{2} y-648 x y^{3}-18 y^{8}+4428 y^{5}+1080 y^{5} x +2484 y^{6}+594 y^{7}-126 y^{10}-315 y^{9}-8 y^{12}-36 y^{11}+72 y^{8} x +216 y^{7} x +594 x y^{6}}
\] |
[_rational] |
✗ |
4.644 |
|
|
\[
{}y^{\prime } = \frac {\left (x y+1\right )^{3}}{x^{5}}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
2.569 |
|
|
\[
{}y^{\prime } = \frac {x \left (-x^{2}+2 x^{2} y-2 x^{4}+1\right )}{y-x^{2}}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
2.139 |
|
|
\[
{}y^{\prime } = y \left (y^{2}+y \,{\mathrm e}^{b x}+{\mathrm e}^{2 b x}\right ) {\mathrm e}^{-2 b x}
\] |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
2.497 |
|
|
\[
{}y^{\prime } = y^{3}-3 y^{2} x^{2}+3 y x^{4}-x^{6}+2 x
\] |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
1.508 |
|
|
\[
{}y^{\prime } = y^{3}+y^{2} x^{2}+\frac {y x^{4}}{3}+\frac {x^{6}}{27}-\frac {2 x}{3}
\] |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
1.559 |
|
|
\[
{}y^{\prime } = \frac {y \left (y^{2} x^{7}+y x^{4}+x -3\right )}{x}
\] |
[_rational, _Abel] |
✗ |
2.322 |
|
|
\[
{}y^{\prime } = y \left (y^{2}+{\mathrm e}^{-x^{2}} y+{\mathrm e}^{-2 x^{2}}\right ) {\mathrm e}^{2 x^{2}} x
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘], _Abel] |
✓ |
6.232 |
|
|
\[
{}y^{\prime } = \frac {y \left (y^{2}+x y+x^{2}+x \right )}{x^{2}}
\] |
[[_homogeneous, ‘class D‘], _rational, _Abel] |
✓ |
1.970 |
|
|
\[
{}y^{\prime } = \frac {y^{3}-3 x y^{2}+3 x^{2} y-x^{3}+x}{x}
\] |
[[_1st_order, _with_linear_symmetries], _rational, _Abel] |
✓ |
1.608 |
|
|
\[
{}y^{\prime } = \frac {x^{3} y^{3}+6 y^{2} x^{2}+12 x y+8+2 x}{x^{3}}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
2.118 |
|
|
\[
{}y^{\prime } = \frac {y^{3} a^{3} x^{3}+3 y^{2} a^{2} x^{2}+3 a x y+1+a^{2} x}{x^{3} a^{3}}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
2.130 |
|
|
\[
{}y^{\prime } = \frac {y \,{\mathrm e}^{-\frac {x^{2}}{2}} \left (2 y^{2}+2 y \,{\mathrm e}^{\frac {x^{2}}{4}}+2 \,{\mathrm e}^{\frac {x^{2}}{2}}+x \,{\mathrm e}^{\frac {x^{2}}{2}}\right )}{2}
\] |
[_Abel] |
✓ |
36.518 |
|
|
\[
{}y^{\prime } = \frac {y^{3}-3 x y^{2}+3 x^{2} y-x^{3}+x^{2}}{\left (x -1\right ) \left (x +1\right )}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
2.885 |
|
|
\[
{}y^{\prime } = \frac {y \left (y^{2} x^{2}+y x \,{\mathrm e}^{x}+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-2 x} \left (x -1\right )}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘], _Abel] |
✗ |
3.990 |
|
|
\[
{}y^{\prime } = \frac {\left (x y+1\right ) \left (y^{2} x^{2}+x^{2} y+2 x y+1+x +x^{2}\right )}{x^{5}}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
2.165 |
|
|
\[
{}y^{\prime } = \frac {y^{3}-3 x y^{2} \ln \left (x \right )+3 x^{2} \ln \left (x \right )^{2} y-x^{3} \ln \left (x \right )^{3}+x^{2}+x y}{x^{2}}
\] |
[_Abel] |
✓ |
2.296 |
|
|
\[
{}y^{\prime } = -F \left (x \right ) \left (-x^{2} a +y^{2}\right )+\frac {y}{x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
2.157 |
|
|
\[
{}y^{\prime } = -F \left (x \right ) \left (-x^{2}-2 x y+y^{2}\right )+\frac {y}{x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
2.802 |
|
|
\[
{}y^{\prime } = -F \left (x \right ) \left (-y^{2} a -b \,x^{2}\right )+\frac {y}{x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
2.420 |
|
|
\[
{}y^{\prime } = -F \left (x \right ) \left (-y^{2}+2 x^{2} y+1-x^{4}\right )+2 x
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
2.749 |
|
|
\[
{}y^{\prime } = -F \left (x \right ) \left (x^{2}+2 x y-y^{2}\right )+\frac {y}{x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
2.751 |
|
|
\[
{}y^{\prime } = -F \left (x \right ) \left (-7 x y^{2}-x^{3}\right )+\frac {y}{x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
2.291 |
|
|
\[
{}y^{\prime } = -F \left (x \right ) \left (-y^{2}-2 y \ln \left (x \right )-\ln \left (x \right )^{2}\right )+\frac {y}{\ln \left (x \right ) x}
\] |
[_Riccati] |
✓ |
2.956 |
|
|
\[
{}y^{\prime } = -x^{3} \left (-y^{2}-2 y \ln \left (x \right )-\ln \left (x \right )^{2}\right )+\frac {y}{\ln \left (x \right ) x}
\] |
[_Riccati] |
✓ |
3.494 |
|
|
\[
{}y^{\prime } = \left (y-{\mathrm e}^{x}\right )^{2}+{\mathrm e}^{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
1.620 |
|
|
\[
{}y^{\prime } = \frac {\left (y-\operatorname {Si}\left (x \right )\right )^{2}+\sin \left (x \right )}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
6.656 |
|
|
\[
{}y^{\prime } = \left (y+\cos \left (x \right )\right )^{2}+\sin \left (x \right )
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
2.275 |
|
|
\[
{}y^{\prime } = \frac {\left (y-\ln \left (x \right )-\operatorname {Ci}\left (x \right )\right )^{2}+\cos \left (x \right )}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
54.532 |
|
|
\[
{}y^{\prime } = \frac {\left (y-x +\ln \left (x +1\right )\right )^{2}+x}{x +1}
\] |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
2.266 |
|
|
\[
{}y^{\prime } = \frac {2 x^{2} y+x^{3}+y \ln \left (x \right ) x -y^{2}-x y}{x^{2} \left (x +\ln \left (x \right )\right )}
\] |
[_Riccati] |
✓ |
2.911 |
|
|
\[
{}y^{\prime \prime } = 0
\] |
[[_2nd_order, _quadrature]] |
✓ |
1.394 |
|
|
\[
{}y^{\prime \prime }+y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.915 |
|
|
\[
{}y^{\prime \prime }+y-\sin \left (n x \right ) = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
2.345 |
|
|
\[
{}y^{\prime \prime }+y-\cos \left (b x \right ) a = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
2.349 |
|
|
\[
{}y^{\prime \prime }+y-\sin \left (a x \right ) \sin \left (b x \right ) = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
12.858 |
|
|
\[
{}y^{\prime \prime }-y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
2.030 |
|
|
\[
{}y^{\prime \prime }-2 y-4 x^{2} {\mathrm e}^{x^{2}} = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.343 |
|
|
\[
{}y^{\prime \prime }+a^{2} y-\cot \left (a x \right ) = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.517 |
|
|
\[
{}y^{\prime \prime }+l y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.691 |
|
|
\[
{}y^{\prime \prime }+\left (a x +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.760 |
|
|
\[
{}y^{\prime \prime }-\left (x^{2}+1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.900 |
|
|
\[
{}y^{\prime \prime }-\left (x^{2}+a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.381 |
|
|
\[
{}y^{\prime \prime }-\left (a^{2} x^{2}+a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.957 |
|
|
\[
{}y^{\prime \prime }-c \,x^{a} y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.950 |
|