Number of problems in this table is 173
|
# |
ODE |
CAS classification |
Solved? |
Verified? |
|
\[ {}\frac {d}{d x}y \left (x \right ) = y \left (x \right )^{\frac {1}{3}} \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right ) = -1+x \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\tan \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right ) = y \left (x \right ) \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = \frac {3 x^{2}}{-4+3 y \left (x \right )^{2}} \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = \frac {-3 x^{2} y \left (x \right )-y \left (x \right )^{2}}{2 x^{3}+3 x y \left (x \right )} \] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = \tan \left (x \right ) \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = \cos \left (x \right )-y \left (x \right ) \tan \left (x \right ) \] |
[_linear] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = {| y \left (x \right )|}+1 \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}x \left (\frac {d}{d x}y \left (x \right )\right )+\left (1+x \cot \left (x \right )\right ) y \left (x \right ) = 0 \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right )+\tan \left (k x \right ) y \left (x \right ) = 0 \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\left (-1+x \right ) \left (\frac {d}{d x}y \left (x \right )\right )+3 y \left (x \right ) = \frac {1+\left (-1+x \right ) \sec \left (x \right )^{2}}{\left (-1+x \right )^{3}} \] |
[_linear] |
✓ |
✓ |
|
|
\[ {}x \left (\frac {d}{d x}y \left (x \right )\right )-2 y \left (x \right ) = -1 \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = a y \left (x \right )-b y \left (x \right )^{2} \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = 3 x \left (y \left (x \right )-1\right )^{\frac {1}{3}} \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = 3 x \left (y \left (x \right )-1\right )^{\frac {1}{3}} \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}-4 y \left (x \right ) \cos \left (x \right )+4 \sin \left (x \right ) \cos \left (x \right )+\sec \left (x \right )^{2}+\left (4 y \left (x \right )-4 \sin \left (x \right )\right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\[ {}\left (y \left (x \right )^{3}-1\right ) {\mathrm e}^{x}+3 y \left (x \right )^{2} \left (1+{\mathrm e}^{x}\right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\sin \left (x \right )-y \left (x \right ) \sin \left (x \right )-2 \cos \left (x \right )+\cos \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[_linear] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right )+\frac {2 y \left (x \right )}{x} = -\frac {2 x y \left (x \right )}{x^{2}+2 x^{2} y \left (x \right )+1} \] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right )-\frac {3 y \left (x \right )}{x} = \frac {2 x^{4} \left (4 x^{3}-3 y \left (x \right )\right )}{3 x^{5}+3 x^{3}+2 y \left (x \right )} \] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right )+2 x y \left (x \right ) = -\frac {{\mathrm e}^{-x^{2}} \left (3 x +2 y \left (x \right ) {\mathrm e}^{x^{2}}\right )}{2 x +3 y \left (x \right ) {\mathrm e}^{x^{2}}} \] |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[ {}\cos \left (y \left (t \right )\right ) y^{\prime }\left (t \right ) = -\frac {t \sin \left (y \left (t \right )\right )}{t^{2}+1} \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}2 t \cos \left (y \left (t \right )\right )+3 t^{2} y \left (t \right )+\left (t^{3}-t^{2} \sin \left (y \left (t \right )\right )-y \left (t \right )\right ) y^{\prime }\left (t \right ) = 0 \] |
[_exact] |
✓ |
✓ |
|
|
\[ {}2 t -2 \,{\mathrm e}^{t y \left (t \right )} \sin \left (2 t \right )+{\mathrm e}^{t y \left (t \right )} \cos \left (2 t \right ) y \left (t \right )+\left (-3+{\mathrm e}^{t y \left (t \right )} t \cos \left (2 t \right )\right ) y^{\prime }\left (t \right ) = 0 \] |
[_exact] |
✓ |
✓ |
|
|
\[ {}y^{\prime }\left (t \right ) = {\mathrm e}^{-t}+\ln \left (1+y \left (t \right )^{2}\right ) \] |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
|
|
\[ {}\sin \left (x \right ) \cos \left (y \left (x \right )\right )+\cos \left (x \right ) \sin \left (y \left (x \right )\right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}x^{2}+y \left (x \right )^{2} = 2 x y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right ) \] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = \frac {y \left (x \right )}{x}+\tan \left (\frac {y \left (x \right )}{x}\right ) \] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
|
|
\[ {}y \left (x \right )-x^{2} \sqrt {x^{2}-y \left (x \right )^{2}}-x \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
|
|
\[ {}y \left (x \right ) \left (x +y \left (x \right )^{2}\right )+x \left (x -y \left (x \right )^{2}\right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
|
|
\[ {}y \left (x \right )+2 \left (x -2 y \left (x \right )^{2}\right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
|
|
\[ {}1+x y \left (x \right ) \left (1+x y \left (x \right )^{2}\right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✗ |
N/A |
|
|
\[ {}x \,{\mathrm e}^{-y \left (x \right )^{2}}+y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\frac {2 y \left (x \right )^{3}-2 x^{2} y \left (x \right )^{3}-x +x y \left (x \right )^{2} \ln \left (y \left (x \right )\right )}{x y \left (x \right )^{2}}+\frac {\left (2 y \left (x \right )^{3} \ln \left (x \right )-x^{2} y \left (x \right )^{3}+2 x +x y \left (x \right )^{2}\right ) \left (\frac {d}{d x}y \left (x \right )\right )}{y \left (x \right )^{3}} = 0 \] |
[_exact] |
✓ |
✓ |
|
|
\[ {}x^{2}+y \left (x \right )^{2} = 2 x y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right ) \] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
|
|
\[ {}y \left (x \right )^{2}+\left (x^{3}-2 x y \left (x \right )\right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = \ln \left (x y \left (x \right )\right ) \] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = \sqrt {1+x y \left (x \right )} \] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = \cos \left (x \right )+\sin \left (y \left (x \right )\right ) \] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[ {}y^{\prime }\left (t \right ) = -\tan \left (t \right ) y \left (t \right )+\sec \left (t \right ) \] |
[_linear] |
✓ |
✓ |
|
|
\[ {}x \left (1-2 x^{2} y \left (x \right )\right ) \left (\frac {d}{d x}y \left (x \right )\right )+y \left (x \right ) = 3 x^{2} y \left (x \right )^{2} \] |
[[_homogeneous, ‘class G‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[ {}\left (2 \sin \left (y \left (x \right )\right )-x \right ) \left (\frac {d}{d x}y \left (x \right )\right ) = \tan \left (y \left (x \right )\right ) \] |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
|
|
\[ {}\left (2 \sin \left (y \left (x \right )\right )-x \right ) \left (\frac {d}{d x}y \left (x \right )\right ) = \tan \left (y \left (x \right )\right ) \] |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = \frac {2 \sqrt {y \left (x \right )-1}}{3} \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}y^{\prime }\left (t \right )-3 y \left (t \right ) = -10 \,{\mathrm e}^{-t +a} \sin \left (-2 t +2 a \right ) \operatorname {Heaviside}\left (t -a \right ) \] |
[[_linear, ‘class A‘]] |
✓ |
✓ |
|
|
\[ {}\cot \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right ) = y \left (x \right ) \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\left (x^{3}+1\right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 3 x^{2} \tan \left (x \right ) \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}x +y \left (x \right )+\left (x -y \left (x \right )\right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\[ {}\cos \left (y \left (x \right )\right )-x \sin \left (y \left (x \right )\right ) \left (\frac {d}{d x}y \left (x \right )\right ) = \sec \left (x \right )^{2} \] |
[_exact] |
✓ |
✓ |
|
|
\[ {}y \left (x \right )^{2} \left (\frac {d}{d x}y \left (x \right )\right ) = 2+3 y \left (x \right )^{6} \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}x^{2}+y \left (x \right )^{2} = 2 x y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right ) \] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right )-\frac {y \left (x \right )}{x}+\csc \left (\frac {y \left (x \right )}{x}\right ) = 0 \] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
|
|
\[ {}{\mathrm e}^{x} \left (y \left (x \right )^{3}+x y \left (x \right )^{3}+1\right )+3 y \left (x \right )^{2} \left (x \,{\mathrm e}^{x}-6\right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[_exact, _Bernoulli] |
✓ |
✓ |
|
|
\[ {}y \left (x \right )^{2} {\mathrm e}^{x y \left (x \right )^{2}}+4 x^{3}+\left (2 x y \left (x \right ) {\mathrm e}^{x y \left (x \right )^{2}}-3 y \left (x \right )^{2}\right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[_exact] |
✓ |
✓ |
|
|
\[ {}\left (x -\cos \left (y \left (x \right )\right )\right ) \left (\frac {d}{d x}y \left (x \right )\right )+\tan \left (y \left (x \right )\right ) = 0 \] |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
|
|
\[ {}\left (\frac {d}{d x}y \left (x \right )\right ) \sin \left (x \right ) = y \left (x \right ) \ln \left (y \left (x \right )\right ) \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}1+y \left (x \right )^{2}+x y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = \frac {2 x y \left (x \right )^{2}+x}{x^{2} y \left (x \right )-y \left (x \right )} \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = \left (1+y \left (x \right )^{2}\right ) \tan \left (x \right ) \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = y \left (x \right )^{\frac {1}{3}} \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}\cos \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )+y \left (x \right ) \sin \left (x \right ) = 2 x \cos \left (x \right )^{2} \] |
[_linear] |
✓ |
✓ |
|
|
\[ {}\cos \left (y \left (x \right )\right )+\left (1+{\mathrm e}^{-x}\right ) \sin \left (y \left (x \right )\right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right )-y \left (x \right ) \cot \left (x \right ) = y \left (x \right )^{2} \sec \left (x \right )^{2} \] |
[_Bernoulli] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right )-y \left (x \right ) \tan \left (x \right ) = \cos \left (x \right )-2 x \sin \left (x \right ) \] |
[_linear] |
✓ |
✓ |
|
|
\[ {}\frac {r \left (\theta \right ) \tan \left (\theta \right ) \left (\frac {d}{d \theta }r \left (\theta \right )\right )}{a^{2}-r \left (\theta \right )^{2}} = 1 \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right )+y \left (x \right ) \cot \left (x \right ) = \cos \left (x \right ) \] |
[_linear] |
✓ |
✓ |
|
|
\[ {}\cos \left (y \left (x \right )\right )+\left (1+{\mathrm e}^{-x}\right ) \sin \left (y \left (x \right )\right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = 3 y \left (x \right )^{\frac {2}{3}} \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}{\mathrm e}^{x}-\left (1+{\mathrm e}^{x}\right ) y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}x^{2}+2 x y \left (x \right )-y \left (x \right )^{2}+\left (y \left (x \right )^{2}+2 x y \left (x \right )-x^{2}\right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = \frac {x}{y \left (x \right )}+\frac {y \left (x \right )}{x} \] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
|
|
\[ {}\left (\frac {d}{d x}y \left (x \right )\right ) \left (\frac {d}{d x}y \left (x \right )+y \left (x \right )\right ) = \left (x +y \left (x \right )\right ) x \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}x \left (\frac {d}{d x}y \left (x \right )\right ) = x +\frac {y \left (x \right )}{2} \] |
[_linear] |
✗ |
N/A |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = y \left (x \right )^{2} \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = 2 \sqrt {y \left (x \right )} \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = 2 \sqrt {y \left (x \right )} \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}x \ln \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )+y \left (x \right ) = 3 x^{3} \] |
[_linear] |
✗ |
N/A |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = \frac {x +y \left (x \right )}{x -y \left (x \right )} \] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\[ {}L i^{\prime }\left (t \right )+R i \left (t \right ) = E_{0} \operatorname {Heaviside}\left (t \right ) \] |
[[_linear, ‘class A‘]] |
✓ |
✓ |
|
|
\[ {}L i^{\prime }\left (t \right )+R i \left (t \right ) = E_{0} \delta \left (t \right ) \] |
[[_linear, ‘class A‘]] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = y \left (x \right )+x \,{\mathrm e}^{y \left (x \right )} \] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = y \left (x \right )+x \,{\mathrm e}^{y \left (x \right )} \] |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = \frac {2 y \left (x \right )}{x} \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}p^{\prime }\left (t \right ) = a p \left (t \right )-b p \left (t \right )^{2} \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}\frac {\left (\frac {d}{d x}y \left (x \right )\right ) y \left (x \right )}{1+\frac {\sqrt {1+\left (\frac {d}{d x}y \left (x \right )\right )^{2}}}{2}} = -x \] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = 2 \sqrt {y \left (x \right )} \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = y \left (x \right )^{\frac {1}{3}} \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}x^{\prime }\left (t \right ) = \frac {t^{2}}{1-x \left (t \right )^{2}} \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = y \left (x \right )^{\frac {1}{3}} \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}3 x^{2} y \left (x \right )^{2}-y \left (x \right )^{3}+2 x +\left (2 x^{3} y \left (x \right )-3 x y \left (x \right )^{2}+1\right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[_exact, _rational] |
✓ |
✓ |
|
|
\[ {}2 y \left (x \right ) \sin \left (x \right ) \cos \left (x \right )+y \left (x \right )^{2} \sin \left (x \right )+\left (\sin \left (x \right )^{2}-2 y \left (x \right ) \cos \left (x \right )\right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[_exact, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[ {}{\mathrm e}^{x} y \left (x \right )+2 \,{\mathrm e}^{x}+y \left (x \right )^{2}+\left ({\mathrm e}^{x}+2 x y \left (x \right )\right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[_exact, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[ {}2 x -5 y \left (x \right )+\left (4 x -y \left (x \right )\right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\[ {}{\mathrm e}^{x} \left (y \left (x \right )-3 \left (1+{\mathrm e}^{x}\right )^{2}\right )+\left (1+{\mathrm e}^{x}\right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[_linear] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right )+y \left (x \right ) = \left \{\begin {array}{cc} {\mathrm e}^{-x} & 0\le x <2 \\ {\mathrm e}^{-2} & 2\le x \end {array}\right . \] |
[[_linear, ‘class A‘]] |
✓ |
✓ |
|
|
\[ {}2 y \left (x \right )^{2}+8+\left (-x^{2}+1\right ) y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}4 x +3 y \left (x \right )+1+\left (1+x +y \left (x \right )\right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\[ {}x^{\prime }\left (t \right ) = k x \left (t \right )-x \left (t \right )^{2} \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}x^{\prime }\left (t \right ) = -x \left (t \right ) \left (k^{2}+x \left (t \right )^{2}\right ) \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right )+\frac {y \left (x \right )}{x} = x^{2} \] |
[_linear] |
✗ |
N/A |
|
|
\[ {}y \left (x \right ) = x \left (\frac {d}{d x}y \left (x \right )\right )+\left (\frac {d}{d x}y \left (x \right )\right )^{2} \] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
|
|
\[ {}y \left (x \right ) = x \left (\frac {d}{d x}y \left (x \right )\right )+\left (\frac {d}{d x}y \left (x \right )\right )^{2} \] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = a y \left (x \right )+b \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = -x \sqrt {1-y \left (x \right )^{2}} \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = -\frac {x}{2}+\frac {\sqrt {x^{2}+4 y \left (x \right )}}{2} \] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = -\frac {3 x^{2}}{2 y \left (x \right )} \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = \frac {\sqrt {y \left (x \right )}}{x} \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = \frac {\sqrt {y \left (x \right )}}{x} \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = 3 x y \left (x \right )^{\frac {1}{3}} \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = 3 x y \left (x \right )^{\frac {1}{3}} \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = \sqrt {\left (y \left (x \right )+2\right ) \left (y \left (x \right )-1\right )} \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = \sqrt {\left (y \left (x \right )+2\right ) \left (y \left (x \right )-1\right )} \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = \frac {y \left (x \right )}{y \left (x \right )-x} \] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = \frac {x y \left (x \right )}{x^{2}+y \left (x \right )^{2}} \] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = x \sqrt {1-y \left (x \right )^{2}} \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = -\frac {x}{2}+\frac {\sqrt {x^{2}+4 y \left (x \right )}}{2} \] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = -\frac {x}{2}+\frac {\sqrt {x^{2}+4 y \left (x \right )}}{2} \] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = -\frac {x}{2}+\frac {\sqrt {x^{2}+4 y \left (x \right )}}{2} \] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = -\frac {x}{2}+\frac {\sqrt {x^{2}+4 y \left (x \right )}}{2} \] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
|
|
\[ {}y^{\prime }\left (t \right ) = \left (y \left (t \right )-3\right ) \left (\sin \left (y \left (t \right )\right ) \sin \left (t \right )+\cos \left (t \right )+1\right ) \] |
[‘x=_G(y,y’)‘] |
✗ |
N/A |
|
|
\[ {}x \left (\frac {d}{d x}y \left (x \right )\right ) = \sin \left (x \right ) \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}x \left (\frac {d}{d x}y \left (x \right )\right ) = \sin \left (x^{2}\right ) \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = \left \{\begin {array}{cc} 0 & x <0 \\ 1 & 0\le x \end {array}\right . \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = 2 \sqrt {y \left (x \right )} \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}\left (y \left (x \right )^{2}-1\right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 4 x y \left (x \right ) \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = \frac {4 x -9}{3 \left (x -3\right )^{\frac {2}{3}}} \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}y^{\prime }\left (t \right )+t^{2} = \frac {1}{y \left (t \right )^{2}} \] |
[_rational] |
✗ |
N/A |
|
|
\[ {}y^{\prime }\left (t \right ) = y \left (t \right )^{\frac {1}{5}} \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}\frac {y^{\prime }\left (t \right )}{t} = \sqrt {y \left (t \right )} \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}y^{\prime }\left (t \right ) = 6 y \left (t \right )^{\frac {2}{3}} \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}y^{\prime }\left (t \right ) = \sin \left (y \left (t \right )\right )-\cos \left (t \right ) \] |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
|
|
\[ {}y^{\prime }\left (t \right ) = \sqrt {y \left (t \right )^{2}-1} \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}y^{\prime }\left (t \right ) = \sqrt {y \left (t \right )^{2}-1} \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}y^{\prime }\left (t \right ) = \sqrt {y \left (t \right )^{2}-1} \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}y^{\prime }\left (t \right ) = \sqrt {25-y \left (t \right )^{2}} \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}y^{\prime }\left (t \right ) = \sqrt {25-y \left (t \right )^{2}} \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}y^{\prime }\left (t \right ) = \sqrt {25-y \left (t \right )^{2}} \] |
[_quadrature] |
✓ |
✓ |
|
|
\[ {}y^{\prime }\left (t \right )+\frac {y \left (t \right )}{\sqrt {-t^{2}+4}} = t \] |
[_linear] |
✓ |
✓ |
|
|
\[ {}y^{\prime }\left (t \right )+y \left (t \right ) f \left (t \right ) = 0 \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}y^{\prime }\left (t \right ) = y \left (t \right ) f \left (t \right ) \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\left (1+{\mathrm e}^{t}\right ) y^{\prime }\left (t \right )+{\mathrm e}^{t} y \left (t \right ) = t \] |
[_linear] |
✓ |
✓ |
|
|
\[ {}1+5 t -y \left (t \right )-\left (t +2 y \left (t \right )\right ) y^{\prime }\left (t \right ) = 0 \] |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\[ {}{\mathrm e}^{y \left (t \right )}-2 t y \left (t \right )+\left (t \,{\mathrm e}^{y \left (t \right )}-t^{2}\right ) y^{\prime }\left (t \right ) = 0 \] |
[_exact] |
✗ |
N/A |
|
|
\[ {}2 t y \left (t \right ) {\mathrm e}^{t^{2}}+2 t \,{\mathrm e}^{-y \left (t \right )}+\left ({\mathrm e}^{t^{2}}-t^{2} {\mathrm e}^{-y \left (t \right )}+1\right ) y^{\prime }\left (t \right ) = 0 \] |
[_exact] |
✓ |
✓ |
|
|
\[ {}\cos \left (t \right )^{2}-\sin \left (t \right )^{2}+y \left (t \right )+\left (\sec \left (y \left (t \right )\right ) \tan \left (y \left (t \right )\right )+t \right ) y^{\prime }\left (t \right ) = 0 \] |
[_exact] |
✓ |
✓ |
|
|
\[ {}\frac {1}{t^{2}+1}-y \left (t \right )^{2}-2 t y \left (t \right ) y^{\prime }\left (t \right ) = 0 \] |
[_exact, _rational, _Bernoulli] |
✓ |
✓ |
|
|
\[ {}-2 x -y \left (x \right ) \cos \left (x y \left (x \right )\right )+\left (2 y \left (x \right )-x \cos \left (x y \left (x \right )\right )\right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[_exact] |
✓ |
✓ |
|
|
\[ {}-4 x^{3}+6 y \left (x \right ) \sin \left (6 x y \left (x \right )\right )+\left (4 y \left (x \right )^{3}+6 x \sin \left (6 x y \left (x \right )\right )\right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[_exact] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right )+y \left (x \right ) \cot \left (x \right ) = y \left (x \right )^{4} \] |
[_Bernoulli] |
✓ |
✗ |
|
|
\[ {}y^{\prime }\left (t \right ) = \frac {y \left (t \right )^{2}-t^{2}}{t y \left (t \right )} \] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
|
|
\[ {}y \left (t \right ) \sin \left (\frac {t}{y \left (t \right )}\right )-\left (t +t \sin \left (\frac {t}{y \left (t \right )}\right )\right ) y^{\prime }\left (t \right ) = 0 \] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
|
|
\[ {}\cos \left (t -y \left (t \right )\right )+\left (1-\cos \left (t -y \left (t \right )\right )\right ) y^{\prime }\left (t \right ) = 0 \] |
[[_homogeneous, ‘class C‘], _exact, _dAlembert] |
✓ |
✓ |
|
|
\[ {}y \left (t \right ) {\mathrm e}^{t y \left (t \right )}-2 t +t \,{\mathrm e}^{t y \left (t \right )} y^{\prime }\left (t \right ) = 0 \] |
[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[ {}y \left (t \right )^{2}+\left (2 t y \left (t \right )-2 \cos \left (y \left (t \right )\right ) \sin \left (y \left (t \right )\right )\right ) y^{\prime }\left (t \right ) = 0 \] |
[_exact, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
|
|
\[ {}\frac {y \left (t \right )}{t}+\ln \left (y \left (t \right )\right )+\left (\frac {t}{y \left (t \right )}+\ln \left (t \right )\right ) y^{\prime }\left (t \right ) = 0 \] |
[_exact] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = \sqrt {x -y \left (x \right )} \] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = x +y \left (x \right )^{\frac {1}{3}} \] |
[_Chini] |
✗ |
N/A |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = \sin \left (x^{2} y \left (x \right )\right ) \] |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
|
|
\[ {}y^{\prime }\left (t \right ) = \frac {t}{y \left (t \right )^{3}} \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}y^{\prime }\left (t \right ) = -\frac {y \left (t \right )}{t -2} \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = \sin \left (x y \left (x \right )\right ) \] |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
|
|
\[ {}x \sqrt {1-y \left (x \right )^{2}}+y \left (x \right ) \sqrt {-x^{2}+1}\, \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}x \left (\frac {d}{d x}y \left (x \right )\right )+y \left (x \right ) = a \left (1+x y \left (x \right )\right ) \] |
[_linear] |
✓ |
✓ |
|
|
\[ {}a^{2}+y \left (x \right )^{2}+2 x \sqrt {x a -x^{2}}\, \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = \frac {y \left (x \right )}{x} \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}\left (x^{2}+1\right ) \left (\frac {d}{d x}y \left (x \right )\right )-\frac {\cos \left (2 y \left (x \right )\right )^{2}}{2} = 0 \] |
[_separable] |
✓ |
✓ |
|
|
\[ {}x^{2} \left (\frac {d}{d x}y \left (x \right )\right )+\sin \left (2 y \left (x \right )\right ) = 1 \] |
[_separable] |
✓ |
✗ |
|
|
\[ {}\frac {d}{d x}y \left (x \right )-y \left (x \right ) = -2 \,{\mathrm e}^{-x} \] |
[[_linear, ‘class A‘]] |
✓ |
✓ |
|
|
\[ {}\cos \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-y \left (x \right ) \sin \left (x \right ) = -\sin \left (2 x \right ) \] |
[_linear] |
✓ |
✓ |
|
|
\[ {}\frac {2 x}{y \left (x \right )^{3}}+\frac {\left (y \left (x \right )^{2}-3 x^{2}\right ) \left (\frac {d}{d x}y \left (x \right )\right )}{y \left (x \right )^{4}} = 0 \] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
|
|
\[ {}1+{\mathrm e}^{\frac {x}{y \left (x \right )}}+{\mathrm e}^{\frac {x}{y \left (x \right )}} \left (1-\frac {x}{y \left (x \right )}\right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] |
[[_homogeneous, ‘class A‘], _exact, _dAlembert] |
✓ |
✓ |
|
|
\[ {}\frac {d^{3}}{d x^{3}}y \left (x \right ) = 3 y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right ) \] |
[[_3rd_order, _missing_x], [_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]] |
✗ |
N/A |
|
|
\[ {}\frac {d}{d x}y \left (x \right ) = {\mathrm e}^{y \left (x \right )}+x y \left (x \right ) \] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
|
||||
|
|
||||