|
# |
ODE |
CAS classification |
Solved? |
|
\[
{}\left (x +1\right )^{2} y^{\prime } = \left (1+y\right )^{2}
\] |
[_separable] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = 1-x^{2}+y^{2}-x^{2} y^{2}
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = 3 x^{2} \left (1+y^{2}\right )
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = \left (4 x +y\right )^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime }+y^{2} = x^{2}+1
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime }+2 x y = 1+x^{2}+y^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = 1+x^{2}+y^{2}+x^{2} y^{2}
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = x^{2}-2 x y+y^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime } = \left (1+y\right )^{2}
\] |
[_separable] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = 1-x^{2}+y^{2}-x^{2} y^{2}
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = 3 x^{2} \left (1+y^{2}\right )
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = \left (4 x +y\right )^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = 1+x^{2}+y^{2}+x^{2} y^{2}
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = x^{2}-2 x y+y^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = 2 \left (x +1\right ) \left (1+y^{2}\right )
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = \frac {x^{2}+x y+y^{2}}{x^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{2}+3 x y+y^{2}-x^{2} y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = t -1-y^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = 1+2 x +y^{2}+2 x y^{2}
\] |
[_separable] |
✓ |
|
|
\[
{}2 y^{\prime }+x \left (y^{2}-1\right ) = 0
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = x^{2} \left (1+y^{2}\right )
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = x \left (1+y^{2}\right )
\] |
[_separable] |
✓ |
|
|
\[
{}\frac {y^{\prime }}{\left (1+y\right )^{2}}-\frac {1}{x \left (1+y\right )} = -\frac {3}{x^{2}}
\] |
[[_homogeneous, ‘class C‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = x^{2} \left (1+y^{2}\right )
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = \left (x -1\right ) \left (-1+y\right ) \left (y-2\right )
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } \left (x^{2}+2\right ) = 4 x \left (y^{2}+2 y+1\right )
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = \frac {1+y^{2}}{x^{2}+1}
\] |
[_separable] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = y^{2}+x y-x^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = y^{2}+x y-x^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = x^{2}+x y+y^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {y^{2}-3 x y-5 x^{2}}{x^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = 2 x^{2}+y^{2}+4 x y
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = y^{2}+x y-4 x^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{3} y^{\prime } = 2 y^{2}+2 x^{2} y-2 x^{4}
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2} {\mathrm e}^{-x}+4 y+2 \,{\mathrm e}^{x}
\] |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {y^{2}+y \tan \left (x \right )+\tan \left (x \right )^{2}}{\sin \left (x \right )^{2}}
\] |
[_Riccati] |
✓ |
|
|
\[
{}x \ln \left (x \right )^{2} y^{\prime } = -4 \ln \left (x \right )^{2}+y \ln \left (x \right )+y^{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = 1+x -\left (2 x +1\right ) y+x y^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}x^{2} \left (y^{\prime }+y^{2}\right )-x \left (x +2\right ) y+x +2 = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime }+y^{2}+4 x y+4 x^{2}+2 = 0
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}\left (2 x +1\right ) \left (y^{\prime }+y^{2}\right )-2 y-2 x -3 = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}\left (3 x -1\right ) \left (y^{\prime }+y^{2}\right )-\left (3 x +2\right ) y-6 x +8 = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} \left (y^{\prime }+y^{2}\right )+x y+x^{2}-\frac {1}{4} = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}x^{2} \left (y^{\prime }+y^{2}\right )-7 x y+7 = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
|
|
\[
{}\left (t^{2}+1\right ) y^{\prime } = 1+y^{2}
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = 1-t +y^{2}-t y^{2}
\] |
[_separable] |
✓ |
|
|
\[
{}\left (t^{2}+1\right ) y^{\prime } = 1+y^{2}
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = 1-t +y^{2}-t y^{2}
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = {\mathrm e}^{t}+y^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+\cos \left (t \right )^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = 1-t +y^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}1+y^{2}+\left (x^{2}+1\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
|
|
\[
{}1+y^{2} = \frac {y^{\prime }}{x^{3} \left (x -1\right )}
\] |
[_separable] |
✓ |
|
|
\[
{}\left (x^{2}+x +1\right ) y^{\prime } = y^{2}+2 y+5
\] |
[_separable] |
✓ |
|
|
\[
{}\left (x^{2}-2 x -8\right ) y^{\prime } = y^{2}+y-2
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime }-\frac {y^{2}}{x^{2}} = {\frac {1}{4}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime }-\frac {y^{2}}{x^{2}} = {\frac {1}{4}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {x \left (y^{2}-1\right )}{2 \left (x -2\right ) \left (x -1\right )}
\] |
[_separable] |
✓ |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime }+y^{2} = -1
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (x +y\right )^{2}}{2 x^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = y^{2}+3 x y+x^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {x \left (y^{2}-1\right )}{2 \left (x -2\right ) \left (x -1\right )}
\] |
[_separable] |
✓ |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime }+y^{2} = -1
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = \frac {x^{2}+x y+y^{2}}{x^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (x +y\right )^{2}}{2 x^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = y^{2}+3 x y+x^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \left (9 x -y\right )^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \left (4 x +y+2\right )^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = 2 x \left (x +y\right )^{2}-1
\] |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
|
|
\[
{}y^{\prime }+\frac {2 y}{x}-y^{2} = -\frac {2}{x^{2}}
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime }+\frac {7 y}{x}-3 y^{2} = \frac {3}{x^{2}}
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = x \left (-1+y\right )+\left (-1+y\right )^{2}
\] |
[[_homogeneous, ‘class C‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {x^{2}+y^{2}}{2 x^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}2 y^{\prime } x = 1-y^{2}
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = {\mathrm e}^{x} \left (1+y^{2}\right )
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = \left (x +y\right )^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}1 = \frac {y}{1-x^{2} y^{2}}+\frac {x y^{\prime }}{1-x^{2} y^{2}}
\] |
[_exact, _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = x^{5}+x^{3} y^{2}+y
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = y+x^{2}+9 y^{2}
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \left (x +1\right )^{2}+\left (4 y+1\right )^{2}+8 x y+1
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}x^{2}+y+y^{2}-y^{\prime } x = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime }+y^{2} = x^{2}+1
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime }+f \left (x \right )^{2} = f^{\prime }\left (x \right )+y^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime }+1-x = y \left (x +y\right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \left (x +y\right )^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \left (x -y\right )^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = 3-3 x +3 y+\left (x -y\right )^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = 2 x -\left (x^{2}+1\right ) y+y^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = x \left (x^{3}+2\right )-\left (2 x^{2}-y\right ) y
\] |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = 1+x \left (-x^{3}+2\right )+\left (2 x^{2}-y\right ) y
\] |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \cos \left (x \right )-\left (\sin \left (x \right )-y\right ) y
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \cos \left (2 x \right )+\left (\sin \left (2 x \right )+y\right ) y
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = f \left (x \right )+x f \left (x \right ) y+y^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \left (3+x -4 y\right )^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \left (1+4 x +9 y\right )^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = 3 a +3 b x +3 b y^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a +b x +c y^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a \,x^{n -1}+b \,x^{2 n}+c y^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = 1+a \left (x -y\right ) y
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = 1-x -x^{3}+\left (2 x^{2}+1\right ) y-x y^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = x \left (2+x^{2} y-y^{2}\right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = x +\left (1-2 x \right ) y-\left (1-x \right ) y^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = x^{n} \left (a +b y^{2}\right )
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = a \,x^{m}+b \,x^{n} y^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime }+4 \csc \left (x \right ) = \left (3-\cot \left (x \right )\right ) y+y^{2} \sin \left (x \right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime }+\tan \left (x \right ) \left (1-y^{2}\right ) = 0
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = \left (a +b y+c y^{2}\right ) f \left (x \right )
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } x +x^{2}+y^{2} = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = x^{2}+y \left (1+y\right )
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{2}-y+y^{\prime } x = x^{{2}/{3}}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = a +b y^{2}
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } x = a \,x^{2}+y+b y^{2}
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = a \,x^{2 n}+\left (n +b y\right ) y
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = a \,x^{n}+b y+c y^{2}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = k +a \,x^{n}+b y+c y^{2}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = x^{3}+\left (2 x^{2}+1\right ) y+x y^{2}
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x +b x +\left (2+y a x \right ) y = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x +\operatorname {a0} +\operatorname {a1} x +\left (\operatorname {a2} +\operatorname {a3} x y\right ) y = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x +a \,x^{2} y^{2}+2 y = b
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x +x^{m}+\frac {\left (n -m \right ) y}{2}+x^{n} y^{2} = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = a \,x^{m}-b y-c \,x^{n} y^{2}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = 2 x -y+a \,x^{n} \left (x -y\right )^{2}
\] |
[[_1st_order, _with_linear_symmetries], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = y+\left (x^{2}-y^{2}\right ) f \left (x \right )
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
|
|
\[
{}2 y^{\prime } x +1 = 4 i x y+y^{2}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}3 y^{\prime } x = 3 x^{{2}/{3}}+\left (1-3 y\right ) y
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime }+x^{2}+x y+y^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = \left (2 x -y+1\right )^{2}
\] |
[[_homogeneous, ‘class C‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = a +b y^{2}
\] |
[_separable] |
✓ |
|
|
\[
{}x^{2} y^{\prime }+a \,x^{2}+b x y+c y^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = a +b \,x^{n}+x^{2} y^{2}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime }+2+x y \left (4+x y\right ) = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime }+2+a x \left (1-x y\right )-x^{2} y^{2} = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = a +b \,x^{n}+c \,x^{2} y^{2}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = a +b x y+c \,x^{2} y^{2}
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = a +b x y+c \,x^{4} y^{2}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime } = 1+y^{2}
\] |
[_separable] |
✓ |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime } = 1-y^{2}
\] |
[_separable] |
✓ |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime } = 1-\left (2 x -y\right ) y
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}\left (a^{2}+x^{2}\right ) y^{\prime } = a^{2}+3 x y-2 y^{2}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}\left (x -a \right )^{2} y^{\prime }+k \left (x +y-a \right )^{2}+y^{2} = 0
\] |
[[_homogeneous, ‘class C‘], _rational, _Riccati] |
✓ |
|
|
\[
{}\left (x -a \right ) \left (x -b \right ) y^{\prime }+k \left (y-a \right ) \left (y-b \right ) = 0
\] |
[_separable] |
✓ |
|
|
\[
{}\left (x -a \right ) \left (x -b \right ) y^{\prime }+k \left (x +y-a \right ) \left (y+x -b \right )+y^{2} = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}2 x^{2} y^{\prime }+1+2 x y-x^{2} y^{2} = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x \left (1-2 x \right ) y^{\prime } = 4 x -\left (1+4 x \right ) y+y^{2}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}2 x \left (1-x \right ) y^{\prime }+x +\left (1-x \right ) y^{2} = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}a \,x^{2} y^{\prime } = x^{2}+y a x +b^{2} y^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}\left (b \,x^{2}+a \right ) y^{\prime } = A +B y^{2}
\] |
[_separable] |
✓ |
|
|
\[
{}x^{3} y^{\prime } = x^{4}+y^{2}
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{3} y^{\prime } = x^{2} \left (-1+y\right )+y^{2}
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{3} y^{\prime }+20+x^{2} y \left (1-x^{2} y\right ) = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{3} y^{\prime }+3+\left (3-2 x \right ) x^{2} y-x^{6} y^{2} = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}x \left (-x^{2}+1\right ) y^{\prime }+x^{2}+\left (-x^{2}+1\right ) y^{2} = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}x \left (c \,x^{2}+b x +a \right ) y^{\prime }+x^{2}-\left (c \,x^{2}+b x +a \right ) y = y^{2}
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
|
|
\[
{}\left (-x^{4}+1\right ) y^{\prime } = 2 x \left (1-y^{2}\right )
\] |
[_separable] |
✓ |
|
|
\[
{}x \left (-x^{3}+1\right ) y^{\prime } = x^{2}+\left (1-2 x y\right ) y
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}\left (c \,x^{2}+b x +a \right )^{2} \left (y^{\prime }+y^{2}\right )+A = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}x \left (-x^{4}+1\right ) y^{\prime } = 2 x \left (x^{2}-y^{2}\right )+\left (-x^{4}+1\right ) y
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{n} y^{\prime } = x^{2 n -1}-y^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}x^{n} y^{\prime }+x^{2 n -2}+y^{2}+\left (-n +1\right ) x^{n -1} = 0
\] |
[_Riccati] |
✓ |
|
|
\[
{}x^{n} y^{\prime } = a^{2} x^{2 n -2}+b^{2} y^{2}
\] |
[[_homogeneous, ‘class G‘], _Riccati] |
✓ |
|
|
\[
{}x^{n} y^{\prime } = x^{n -1} \left (a \,x^{2 n}+n y-b y^{2}\right )
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } \sqrt {-x^{2}+1} = 1+y^{2}
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } x -a y+y^{2} = x^{-2 a}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x -a y+y^{2} = x^{-\frac {2 a}{3}}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}u^{\prime }+u^{2} = \frac {c}{x^{{4}/{3}}}
\] |
[_rational, [_Riccati, _special]] |
✓ |
|
|
\[
{}u^{\prime }+b u^{2} = \frac {c}{x^{4}}
\] |
[_rational, [_Riccati, _special]] |
✓ |
|
|
\[
{}u^{\prime }-u^{2} = \frac {2}{x^{{8}/{3}}}
\] |
[_rational, [_Riccati, _special]] |
✓ |
|
|
\[
{}y^{\prime } = x^{3}+\frac {2 y}{x}-\frac {y^{2}}{x}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = 2 \tan \left (x \right ) \sec \left (x \right )-y^{2} \sin \left (x \right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {1}{x^{2}}-\frac {y}{x}-y^{2}
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = 1+\frac {y}{x}-\frac {y^{2}}{x^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \left (x +y\right )^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x -y^{2}+1 = 0
\] |
[_separable] |
✓ |
|
|
\[
{}x^{2} y^{\prime }+x^{2}+x y+y^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {1+y^{2}}{x^{2}+1}
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = x y^{2}-\frac {2 y}{x}-\frac {1}{x^{3}}
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {2 y^{2}}{x}+\frac {y}{x}-2 x
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2} {\mathrm e}^{-x}+y-{\mathrm e}^{x}
\] |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \left (1+y^{2}\right ) \tan \left (x \right )
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = \sqrt {1+\sin \left (x \right )}\, \left (1+y^{2}\right )
\] |
[_separable] |
✓ |
|
|
\[
{}\left (x +1\right )^{2} y^{\prime } = 1+y^{2}
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime }+x +x y^{2} = 0
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = -2 \left (2 x +3 y\right )^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}1+y^{2} = \left (x^{2}+x \right ) y^{\prime }
\] |
[_separable] |
✓ |
|
|
\[
{}y^{2}-1+y^{\prime } x = 0
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = x \left (y-4\right )^{2}-2
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (2 y+3\right )^{2}}{\left (4 x +5\right )^{2}}
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = \frac {y^{2}-1}{x^{2}-1}
\] |
[_separable] |
✓ |
|
|
\[
{}\left (x^{4}+1\right ) y^{\prime }+x \left (1+4 y^{2}\right ) = 0
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = \left (x +y+1\right )^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}x^{2}+x y+y^{2} = x^{2} y^{\prime }
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = x^{2} \left (1+y^{2}\right )
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = -\frac {2}{t}+\frac {y}{t}+\frac {y^{2}}{t}
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = -\frac {y}{t}-1-y^{2}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}\left (\phi ^{\prime }-\frac {\phi ^{2}}{2}\right ) \sin \left (\theta \right )^{2}-\phi \sin \left (\theta \right ) \cos \left (\theta \right ) = \frac {\cos \left (2 \theta \right )}{2}+1
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}-y+y^{\prime } x = x^{2}+y^{2}
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = x^{2} y^{2}-4 x^{2}
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = \frac {x^{2}+x y+y^{2}}{x^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (x +y-1\right )^{2}}{2 \left (x +2\right )^{2}}
\] |
[[_homogeneous, ‘class C‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = y+x^{2}+y^{2}
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
|
|
\[
{}1+y^{2}+\left (x^{2}+1\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
|
|
\[
{}\frac {y}{1-x^{2} y^{2}}+\frac {x y^{\prime }}{1-x^{2} y^{2}} = 1
\] |
[_exact, _rational, _Riccati] |
✓ |
|
|
\[
{}\frac {y^{\prime } x +y}{1-x^{2} y^{2}}+x = 0
\] |
[_exact, _rational, _Riccati] |
✓ |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime }+y^{2} = -1
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = \frac {5 x^{2}-x y+y^{2}}{x^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x -2 y+b y^{2} = c \,x^{4}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{2}-y+y^{\prime } x = x^{{2}/{3}}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}u^{\prime }+u^{2} = \frac {1}{x^{{4}/{5}}}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = x^{2}+y^{2}-1
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime }-y^{2}-x -x^{2} = 0
\] |
[_Riccati] |
✓ |
|
|
\[
{}c y^{\prime } = \frac {a x +b y^{2}}{r x}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}c y^{\prime } = \frac {a x +b y^{2}}{r \,x^{2}}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \sin \left (x \right )+y^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = x +y+b y^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime }+y^{2}-a x -b = 0
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime }+y^{2}-2 x^{2} y+x^{4}-2 x -1 = 0
\] |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
|
|
\[
{}y^{\prime }+y^{2}+\left (x y-1\right ) f \left (x \right ) = 0
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime }-y^{2}-x y-x +1 = 0
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime }-\left (x +y\right )^{2} = 0
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime }-y^{2}+\left (x^{2}+1\right ) y-2 x = 0
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime }-y^{2}+\sin \left (x \right ) y-\cos \left (x \right ) = 0
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime }-y^{2}-y \sin \left (2 x \right )-\cos \left (2 x \right ) = 0
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime }+a y^{2}-b \,x^{2 \nu }-c \,x^{\nu -1} = 0
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime }+a y \left (y-x \right )-1 = 0
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime }+x y^{2}-x^{3} y-2 x = 0
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime }+x^{-a -1} y^{2}-x^{a} = 0
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime }-a \,x^{n} \left (1+y^{2}\right ) = 0
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime }+y^{2} \sin \left (x \right )-\frac {2 \sin \left (x \right )}{\cos \left (x \right )^{2}} = 0
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime }+f \left (x \right ) \left (y^{2}+2 a y+b \right ) = 0
\] |
[_separable] |
✓ |
|
|
\[
{}2 y^{\prime }-3 y^{2}-4 a y-b -c \,{\mathrm e}^{-2 a x} = 0
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } x +x^{2}+y^{2} = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x -y^{2}+1 = 0
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } x +a y^{2}-y+b \,x^{2} = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x +a y^{2}-b y+c \,x^{2 b} = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x +a y^{2}-b y-c \,x^{\beta } = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x +x y^{2}-y-a \,x^{3} = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x +x y^{2}-\left (2 x^{2}+1\right ) y-x^{3} = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x +a x y^{2}+2 y+b x = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x +a x y^{2}+b y+c x +d = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x +x^{a} y^{2}+\frac {\left (a -b \right ) y}{2}+x^{b} = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x +a \,x^{\alpha } y^{2}+b y-c \,x^{\beta } = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x +f \left (x \right ) \left (y^{2}-x^{2}\right )-y = 0
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime }+x^{2}+x y+y^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime }-y^{2}-x y-x^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} \left (y^{\prime }+y^{2}\right )+a \,x^{k}-b \left (b -1\right ) = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} \left (y^{\prime }+y^{2}\right )+4 x y+2 = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} \left (y^{\prime }+y^{2}\right )+y a x +b = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} \left (y^{\prime }-y^{2}\right )-a \,x^{2} y+a x +2 = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} \left (y^{\prime }+a y^{2}\right )+b \,x^{\alpha }+c = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime }+y^{2}-2 x y+1 = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}\left (x -a \right ) \left (x -b \right ) y^{\prime }+k \left (x +y-a \right ) \left (y+x -b \right )+y^{2} = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}2 x^{2} y^{\prime }-2 y^{2}-x y+2 a^{2} x = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}2 x^{2} y^{\prime }-2 y^{2}-3 x y+2 a^{2} x = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}x \left (2 x -1\right ) y^{\prime }+y^{2}-\left (1+4 x \right ) y+4 x = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}2 x \left (x -1\right ) y^{\prime }+\left (x -1\right ) y^{2}-x = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}3 x^{2} y^{\prime }-7 y^{2}-3 x y-x^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}3 \left (x^{2}-4\right ) y^{\prime }+y^{2}-x y-3 = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}x^{3} y^{\prime }-y^{2}-x^{4} = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{3} y^{\prime }-x^{4} y^{2}+x^{2} y+20 = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{3} y^{\prime }-x^{6} y^{2}-\left (2 x -3\right ) x^{2} y+3 = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}x \left (x^{2}-1\right ) y^{\prime }+\left (x^{2}-1\right ) y^{2}-x^{2} = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}2 x \left (x^{2}-1\right ) y^{\prime }+2 \left (x^{2}-1\right ) y^{2}-\left (3 x^{2}-5\right ) y+x^{2}-3 = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}3 x \left (x^{2}-1\right ) y^{\prime }+x y^{2}-\left (x^{2}+1\right ) y-3 x = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}\left (a \,x^{2}+b x +c \right ) \left (-y+y^{\prime } x \right )-y^{2}+x^{2} = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x \left (x^{3}-1\right ) y^{\prime }-2 x y^{2}+y+x^{2} = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}\left (a \,x^{2}+b x +c \right )^{2} \left (y^{\prime }+y^{2}\right )+A = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}x^{n} y^{\prime }+y^{2}-\left (n -1\right ) x^{n -1} y+x^{2 n -2} = 0
\] |
[[_homogeneous, ‘class G‘], _Riccati] |
✓ |
|
|
\[
{}x^{n} y^{\prime }-a y^{2}-b \,x^{2 n -2} = 0
\] |
[[_homogeneous, ‘class G‘], _Riccati] |
✓ |
|
|
\[
{}x y^{\prime } \ln \left (x \right )-y^{2} \ln \left (x \right )-\left (2 \ln \left (x \right )^{2}+1\right ) y-\ln \left (x \right )^{3} = 0
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } \sin \left (x \right )-y^{2} \sin \left (x \right )^{2}+\left (\cos \left (x \right )-3 \sin \left (x \right )\right ) y+4 = 0
\] |
[_Riccati] |
✓ |
|
|
\[
{}2 f \left (x \right ) y^{\prime }+2 f \left (x \right ) y^{2}-f^{\prime }\left (x \right ) y-2 f \left (x \right )^{2} = 0
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (-1+y \ln \left (x \right )\right )^{2}}{x}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (-1+2 y \ln \left (x \right )\right )^{2}}{x}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {y+x^{3} a \,{\mathrm e}^{x}+a \,x^{4}+a \,x^{3}-x y^{2} {\mathrm e}^{x}-x^{2} y^{2}-x y^{2}}{x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {y+x^{3} a \ln \left (x +1\right )+a \,x^{4}+a \,x^{3}-x y^{2} \ln \left (x +1\right )-x^{2} y^{2}-x y^{2}}{x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {y+x^{3} \ln \left (x \right )+x^{4}+x^{3}+7 x y^{2} \ln \left (x \right )+7 x^{2} y^{2}+7 x y^{2}}{x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {y+x^{3} b \ln \left (\frac {1}{x}\right )+x^{4} b +b \,x^{3}+x a y^{2} \ln \left (\frac {1}{x}\right )+a \,x^{2} y^{2}+a x y^{2}}{x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {y+\ln \left (\left (x -1\right ) \left (x +1\right )\right ) x^{3}+7 \ln \left (\left (x -1\right ) \left (x +1\right )\right ) x y^{2}}{x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {y-\ln \left (\frac {x +1}{x -1}\right ) x^{3}+\ln \left (\frac {x +1}{x -1}\right ) x y^{2}}{x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {y+{\mathrm e}^{\frac {x +1}{x -1}} x^{3}+{\mathrm e}^{\frac {x +1}{x -1}} x y^{2}}{x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {x y-y-{\mathrm e}^{x +1} x^{3}+{\mathrm e}^{x +1} x y^{2}}{\left (x -1\right ) x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {y \ln \left (x -1\right )+x^{4}+x^{3}+x^{2} y^{2}+x y^{2}}{\ln \left (x -1\right ) x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {y \ln \left (x -1\right )+{\mathrm e}^{x +1} x^{3}+7 \,{\mathrm e}^{x +1} x y^{2}}{\ln \left (x -1\right ) x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {-y \,{\mathrm e}^{x}+x y-x^{3} \ln \left (x \right )-x^{3}-x y^{2} \ln \left (x \right )-x y^{2}}{\left (-{\mathrm e}^{x}+x \right ) x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {y \ln \left (x \right ) x -y+2 x^{5} b +2 x^{3} a y^{2}}{\left (x \ln \left (x \right )-1\right ) x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (18 x^{{3}/{2}}+36 y^{2}-12 x^{3} y+x^{6}\right ) \sqrt {x}}{36}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {2 x^{2}+2 x +x^{4}-2 x^{2} y-1+y^{2}}{x +1}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {y \ln \left (x \right )+\cosh \left (x \right ) x a y^{2}+\cosh \left (x \right ) x^{3} b}{x \ln \left (x \right )}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {x +y+y^{2}-2 y \ln \left (x \right ) x +x^{2} \ln \left (x \right )^{2}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (4 \,{\mathrm e}^{-x^{2}}-4 x^{2} {\mathrm e}^{-x^{2}}+4 y^{2}-4 x^{2} {\mathrm e}^{-x^{2}} y+x^{4} {\mathrm e}^{-2 x^{2}}\right ) x}{4}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {30 x^{3}+25 \sqrt {x}+25 y^{2}-20 x^{3} y-100 y \sqrt {x}+4 x^{6}+40 x^{{7}/{2}}+100 x}{25 x}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {y+x^{2} \ln \left (x \right )^{3}+2 x^{2} \ln \left (x \right )^{2} y+x^{2} \ln \left (x \right ) y^{2}}{x \ln \left (x \right )}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {y+x^{3} \ln \left (x \right )^{3}+2 x^{3} \ln \left (x \right )^{2} y+x^{3} \ln \left (x \right ) y^{2}}{x \ln \left (x \right )}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {2 x y^{2}+4 y \ln \left (2 x +1\right ) x +2 \ln \left (2 x +1\right )^{2} x +y^{2}-2+\ln \left (2 x +1\right )^{2}+2 y \ln \left (2 x +1\right )}{2 x +1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {-2 \cos \left (x \right ) x +2 \sin \left (x \right ) x^{2}+2 x +2 y^{2}+4 y \cos \left (x \right ) x -4 x y+x^{2} \cos \left (2 x \right )+3 x^{2}-4 x^{2} \cos \left (x \right )}{2 x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {2 x^{2} \cos \left (x \right )+2 \sin \left (x \right ) x^{3}-2 x \sin \left (x \right )+2 x +2 x^{2} y^{2}-4 y \sin \left (x \right ) x +4 y \cos \left (x \right ) x^{2}+4 x y+3-\cos \left (2 x \right )-2 \sin \left (2 x \right ) x -4 \sin \left (x \right )+x^{2} \cos \left (2 x \right )+x^{2}+4 \cos \left (x \right ) x}{2 x^{3}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = -F \left (x \right ) \left (-a \,x^{2}+y^{2}\right )+\frac {y}{x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = -F \left (x \right ) \left (-x^{2}-2 x y+y^{2}\right )+\frac {y}{x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = -F \left (x \right ) \left (-a y^{2}-b \,x^{2}\right )+\frac {y}{x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = -F \left (x \right ) \left (-y^{2}+2 x^{2} y+1-x^{4}\right )+2 x
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = -F \left (x \right ) \left (x^{2}+2 x y-y^{2}\right )+\frac {y}{x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = -F \left (x \right ) \left (-7 x y^{2}-x^{3}\right )+\frac {y}{x}
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = -F \left (x \right ) \left (-y^{2}-2 y \ln \left (x \right )-\ln \left (x \right )^{2}\right )+\frac {y}{\ln \left (x \right ) x}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = -x^{3} \left (-y^{2}-2 y \ln \left (x \right )-\ln \left (x \right )^{2}\right )+\frac {y}{\ln \left (x \right ) x}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \left (y-{\mathrm e}^{x}\right )^{2}+{\mathrm e}^{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (y-\operatorname {Si}\left (x \right )\right )^{2}+\sin \left (x \right )}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \left (y+\cos \left (x \right )\right )^{2}+\sin \left (x \right )
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (y-\ln \left (x \right )-\operatorname {Ci}\left (x \right )\right )^{2}+\cos \left (x \right )}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (y-x +\ln \left (x +1\right )\right )^{2}+x}{x +1}
\] |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {2 x^{2} y+x^{3}+y \ln \left (x \right ) x -y^{2}-x y}{x^{2} \left (x +\ln \left (x \right )\right )}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a y^{2}+b x +c
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}-a^{2} x^{2}+3 a
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a^{2} x^{2}+b x +c
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a y^{2}+b \,x^{2 n}+c \,x^{n -1}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}+b \,x^{-n -2}
\] |
[[_homogeneous, ‘class G‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}+b \,x^{m}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \left (a \,x^{2 n}+b \,x^{n -1}\right ) y^{2}+c
\] |
[_Riccati] |
✓ |
|
|
\[
{}\left (a_{2} x +b_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+a_{0} x +b_{0} = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = x^{2} y^{2}-a^{2} x^{4}+a \left (1-2 b \right ) x^{2}-b \left (b +1\right )
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = a \,x^{2} y^{2}+b \,x^{n}+c
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}\left (c_{2} x^{2}+b_{2} x +a_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+a_{0} = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}a \,x^{2} \left (x -1\right )^{2} \left (y^{\prime }+\lambda y^{2}\right )+b \,x^{2}+c x +s = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}\left (a \,x^{2}+b x +c \right )^{2} \left (y^{\prime }+y^{2}\right )+A = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}x^{n +1} y^{\prime } = a \,x^{2 n} y^{2}+c \,x^{m}+d
\] |
[_Riccati] |
✓ |
|
|
\[
{}\left (a \,x^{n}+b \right ) y^{\prime } = b y^{2}+a \,x^{n -2}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}\left (a \,x^{n}+b \,x^{m}+c \right ) \left (y^{\prime }-y^{2}\right )+a n \left (n -1\right ) x^{n -2}+b m \left (m -1\right ) x^{m -2} = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a y^{2}+b y+c x +k
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a \,x^{n} y+a \,x^{n -1}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a \,x^{n} y+b \,x^{n -1}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+\left (\alpha x +\beta \right ) y+a \,x^{2}+b x +c
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}-a \,x^{n} \left (b \,x^{m}+c \right ) y+b m \,x^{m -1}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = -a n \,x^{n -1} y^{2}+c \,x^{m} \left (a \,x^{n}+b \right ) y-c \,x^{m}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = a y^{2}+b y+c \,x^{2 b}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = a y^{2}+b y+c \,x^{n}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = a y^{2}+\left (n +b \,x^{n}\right ) y+c \,x^{2 n}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = x y^{2}+a y+b \,x^{n}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x +a_{3} x y^{2}+a_{2} y+a_{1} x +a_{0} = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = a \,x^{n} y^{2}+b y+c \,x^{-n}
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = a \,x^{n} y^{2}+m y-a \,b^{2} x^{n +2 m}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = x^{2 n} y^{2}+\left (m -n \right ) y+x^{2 m}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = a \,x^{n} y^{2}+b y+c \,x^{m}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = a \,x^{2 n} y^{2}+\left (b \,x^{n}-n \right ) y+c
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = a \,x^{2 n +m} y^{2}+\left (b \,x^{n +m}-n \right ) y+c \,x^{m}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}\left (a_{2} x +b_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (a_{1} x +b_{1} \right ) y+a_{0} x +b_{0} = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}\left (a x +c \right ) y^{\prime } = \alpha \left (b x +a y\right )^{2}+\beta \left (b x +a y\right )-b x +\gamma
\] |
[[_1st_order, _with_linear_symmetries], _rational, _Riccati] |
✓ |
|
|
\[
{}2 x^{2} y^{\prime } = 2 y^{2}+x y-2 a^{2} x
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}2 x^{2} y^{\prime } = 2 y^{2}+3 x y-2 a^{2} x
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = a \,x^{2} y^{2}+b x y+c
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = c \,x^{2} y^{2}+\left (a \,x^{2}+b x \right ) y+\alpha \,x^{2}+\beta x +\gamma
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = a \,x^{2} y^{2}+b x y+c \,x^{n}+s
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = a \,x^{2} y^{2}+b x y+c \,x^{2 n}+s \,x^{n}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = c \,x^{2} y^{2}+\left (a \,x^{n}+b \right ) x y+\alpha \,x^{2 n}+\beta \,x^{n}+\gamma
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}\left (a \,x^{2}+b \right ) y^{\prime }+\alpha y^{2}+\beta x y+\frac {b \left (a +\beta \right )}{\alpha } = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}\left (a \,x^{2}+b \right ) y^{\prime }+\alpha y^{2}+\beta x y+\gamma = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}\left (a \,x^{2}+b \right ) y^{\prime }+y^{2}-2 x y+\left (1-a \right ) x^{2}-b = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}\left (a \,x^{2}+b x +c \right ) y^{\prime } = y^{2}+\left (2 \lambda x +b \right ) y+\lambda \left (\lambda -a \right ) x^{2}+\mu
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}\left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime } = y^{2}+\left (a_{1} x +b_{1} \right ) y-\lambda \left (\lambda +a_{1} -a_{2} \right ) x^{2}+\lambda \left (b_{2} -b_{1} \right ) x +\lambda c_{2}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}\left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime } = y^{2}+\left (a_{1} x +b_{1} \right ) y+a_{0} x^{2}+b_{0} x +c_{0}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}\left (x -a \right ) \left (x -b \right ) y^{\prime }+k \left (x +y-a \right ) \left (y+x -b \right )+y^{2} = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}\left (c_{2} x^{2}+b_{2} x +a_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (b_{1} x +a_{1} \right ) y+a_{0} = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}x^{3} y^{\prime } = x^{3} a y^{2}+\left (b \,x^{2}+c \right ) y+s x
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}x^{3} y^{\prime } = x^{3} a y^{2}+x \left (b x +c \right ) y+\alpha x +\beta
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}x \left (x^{2}+a \right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (b \,x^{2}+c \right ) y+s x = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} \left (x +a \right ) \left (y^{\prime }+\lambda y^{2}\right )+x \left (b x +c \right ) y+\alpha x +\beta = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}\left (a \,x^{2}+b x +e \right ) \left (-y+y^{\prime } x \right )-y^{2}+x^{2} = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} \left (x^{2}+a \right ) \left (y^{\prime }+\lambda y^{2}\right )+x \left (b \,x^{2}+c \right ) y+s = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}x^{n +1} y^{\prime } = a \,x^{2 n} y^{2}+b \,x^{n} y+c \,x^{m}+d
\] |
[_Riccati] |
✓ |
|
|
\[
{}x \left (a \,x^{k}+b \right ) y^{\prime } = \alpha \,x^{n} y^{2}+\left (\beta -a n \,x^{k}\right ) y+\gamma \,x^{-n}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} \left (a \,x^{n}-1\right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (p \,x^{n}+q \right ) x y+r \,x^{n}+s = 0
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}\left (a \,x^{n}+b \,x^{m}+c \right ) \left (-y+y^{\prime } x \right )+s \,x^{k} \left (y^{2}-\lambda \,x^{2}\right ) = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a y^{2}+b \,{\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} {\mathrm e}^{2 \lambda x}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \sigma y^{2}+a +b \,{\mathrm e}^{\lambda x}+c \,{\mathrm e}^{2 \lambda x}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \sigma y^{2}+a y+b \,{\mathrm e}^{x}+c
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+b y+a \left (\lambda -b \right ) {\mathrm e}^{\lambda x}-a^{2} {\mathrm e}^{2 \lambda x}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a \,{\mathrm e}^{\lambda x} y-a b \,{\mathrm e}^{\lambda x}-b^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a \,{\mathrm e}^{2 \lambda x} \left ({\mathrm e}^{\lambda x}+b \right )^{n}-\frac {\lambda ^{2}}{4}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a \,{\mathrm e}^{8 \lambda x}+b \,{\mathrm e}^{6 \lambda x}+c \,{\mathrm e}^{4 \lambda x}-\lambda ^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{k x} y^{2}+b \,{\mathrm e}^{s x}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{\lambda x} y^{2}+b y+c \,{\mathrm e}^{-\lambda x}
\] |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{\mu x} y^{2}+\lambda y-a \,b^{2} {\mathrm e}^{\left (\mu +2 \lambda \right ) x}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = {\mathrm e}^{\lambda x} y^{2}+a \,{\mathrm e}^{\mu x} y+a \lambda \,{\mathrm e}^{\left (\mu -\lambda \right ) x}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = -\lambda \,{\mathrm e}^{\lambda x} y^{2}+a \,{\mathrm e}^{\mu x} y-a \,{\mathrm e}^{\left (\mu -\lambda \right ) x}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{k x} y^{2}+b y+c \,{\mathrm e}^{s x}+d \,{\mathrm e}^{-k x}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{\left (\mu +2 \lambda \right ) x} y^{2}+\left (b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}-\lambda \right ) y+c \,{\mathrm e}^{\mu x}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = {\mathrm e}^{\mu x} \left (y-b \,{\mathrm e}^{\lambda x}\right )^{2}+b \lambda \,{\mathrm e}^{\lambda x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}\left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+c \right ) \left (y^{\prime }-y^{2}\right )+a \,\lambda ^{2} {\mathrm e}^{\lambda x}+b \,\mu ^{2} {\mathrm e}^{\mu x} = 0
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a x \,{\mathrm e}^{\lambda x} y+a \,{\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{\lambda x} y^{2}+b \,{\mathrm e}^{-\lambda x}
\] |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = -\lambda \,{\mathrm e}^{\lambda x} y^{2}+a \,x^{n} {\mathrm e}^{\lambda x} y-a \,x^{n}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{\lambda x} y^{2}-a b \,x^{n} {\mathrm e}^{\lambda x} y+b n \,x^{n -1}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}+\lambda y-a \,b^{2} x^{n} {\mathrm e}^{2 \lambda x}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+a \,x^{k +1} {\mathrm e}^{\lambda x} y-a \,{\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a \,x^{n} {\mathrm e}^{2 \lambda x} y^{2}+\left (b \,x^{n} {\mathrm e}^{\lambda x}-\lambda \right ) y+c \,x^{n}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{\lambda x} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = a \,{\mathrm e}^{\lambda x} y^{2}+k y+a \,b^{2} x^{2 k} {\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{-\lambda \,x^{2}} y^{2}+\lambda x y+a \,b^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}+\lambda x y+a \,b^{2} x^{n} {\mathrm e}^{\lambda \,x^{2}}
\] |
[_Riccati] |
✓ |
|
|
\[
{}x^{4} \left (y^{\prime }-y^{2}\right ) = a +b \,{\mathrm e}^{\frac {k}{x}}+c \,{\mathrm e}^{\frac {2 k}{x}}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}-a^{2}+a \lambda \sinh \left (\lambda x \right )-a^{2} \sinh \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a \sinh \left (\beta x \right ) y+a b \sinh \left (\beta x \right )-b^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a x \sinh \left (b x \right )^{m} y+a \sinh \left (b x \right )^{m}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \lambda \sinh \left (\lambda x \right ) y^{2}-\lambda \sinh \left (\lambda x \right )^{3}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \left (a \sinh \left (\lambda x \right )^{2}-\lambda \right ) y^{2}-a \sinh \left (\lambda x \right )^{2}+\lambda -a
\] |
[_Riccati] |
✓ |
|
|
\[
{}\left (a \sinh \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )+a \,\lambda ^{2} \sinh \left (\lambda x \right ) = 0
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \alpha y^{2}+\beta +\gamma \cosh \left (x \right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a \cosh \left (\beta x \right ) y+a b \cosh \left (\beta x \right )-b^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a x \cosh \left (b x \right )^{m} y+a \cosh \left (b x \right )^{m}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \left (a \cosh \left (\lambda x \right )^{2}-\lambda \right ) y^{2}+a +\lambda -a \cosh \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}2 y^{\prime } = \left (a -\lambda +a \cosh \left (\lambda x \right )\right ) y^{2}+a +\lambda -a \cosh \left (\lambda x \right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a \sinh \left (\lambda x \right ) y^{2}+b \sinh \left (\lambda x \right ) \cosh \left (\lambda x \right )^{n}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a \cosh \left (\lambda x \right ) y^{2}+b \cosh \left (\lambda x \right ) \sinh \left (\lambda x \right )^{n}
\] |
[_Riccati] |
✓ |
|
|
\[
{}\left (a \cosh \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )+a \,\lambda ^{2} \cosh \left (\lambda x \right ) = 0
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a \lambda -a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+3 a \lambda -\lambda ^{2}-a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a x \tanh \left (b x \right )^{m} y+a \tanh \left (b x \right )^{m}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a \lambda -a \left (a +\lambda \right ) \coth \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}-\lambda ^{2}+3 a \lambda -a \left (a +\lambda \right ) \coth \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a x \coth \left (b x \right )^{m} y+a \coth \left (b x \right )^{m}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}-2 \lambda ^{2} \tanh \left (\lambda x \right )^{2}-2 \lambda ^{2} \coth \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a \lambda +b \lambda -2 a b -a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}-b \left (b +\lambda \right ) \coth \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = a y^{2}+b \ln \left (x \right )+c
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = a y^{2}+b \ln \left (x \right )^{k}+c \ln \left (x \right )^{2 k +2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = x^{2} y^{2}+a \ln \left (x \right )^{2}+b \ln \left (x \right )+c
\] |
[_Riccati] |
✓ |
|
|
\[
{}x^{2} \ln \left (a x \right ) \left (y^{\prime }-y^{2}\right ) = 1
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a \ln \left (\beta x \right ) y-a b \ln \left (\beta x \right )-b^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a x \ln \left (b x \right )^{m} y+a \ln \left (b x \right )^{m}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}-a b \,x^{n +1} \ln \left (x \right ) y+b \ln \left (x \right )+b
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = -\left (n +1\right ) x^{n} y^{2}+a \,x^{n +1} \ln \left (x \right )^{m} y-a \ln \left (x \right )^{m}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a \ln \left (x \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = \left (a y+b \ln \left (x \right )\right )^{2}
\] |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = a \ln \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \ln \left (\lambda x \right )^{m}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = a \,x^{n} \left (y+b \ln \left (x \right )\right )^{2}-b
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = a \,x^{2 n} \ln \left (x \right ) y^{2}+\left (b \,x^{n} \ln \left (x \right )-n \right ) y+c \ln \left (x \right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = a^{2} x^{2} y^{2}-x y+b^{2} \ln \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \alpha y^{2}+\beta +\gamma \sin \left (\lambda x \right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}-a^{2}+a \lambda \sin \left (\lambda x \right )+a^{2} \sin \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a \sin \left (\beta x \right ) y+a b \sin \left (\beta x \right )-b^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \lambda \sin \left (\lambda x \right ) y^{2}+\lambda \sin \left (\lambda x \right )^{3}
\] |
[_Riccati] |
✓ |
|
|
\[
{}2 y^{\prime } = \left (\lambda +a -a \sin \left (\lambda x \right )\right ) y^{2}+\lambda -a -a \sin \left (\lambda x \right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \left (\lambda +a \sin \left (\lambda x \right )^{2}\right ) y^{2}+\lambda -a +a \sin \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+a \,x^{k +1} \sin \left (x \right )^{m} y-a \sin \left (x \right )^{m}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a \sin \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = a \sin \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \sin \left (\lambda x \right )^{m}
\] |
[_Riccati] |
✓ |
|
|
\[
{}\left (a \sin \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )-a \,\lambda ^{2} \sin \left (\lambda x \right ) = 0
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \alpha y^{2}+\beta +\gamma \cos \left (\lambda x \right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}-a^{2}+a \lambda \cos \left (\lambda x \right )+a^{2} \cos \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a \cos \left (\beta x \right ) y+a b \cos \left (\beta x \right )-b^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \lambda \cos \left (\lambda x \right ) y^{2}+\lambda \cos \left (\lambda x \right )^{3}
\] |
[_Riccati] |
✓ |
|
|
\[
{}2 y^{\prime } = \left (\lambda +a -a \cos \left (\lambda x \right )\right ) y^{2}+\lambda -a -a \cos \left (\lambda x \right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \left (\lambda +a \cos \left (\lambda x \right )^{2}\right ) y^{2}+\lambda -a +a \cos \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+a \,x^{k +1} \cos \left (x \right )^{m} y-a \cos \left (x \right )^{m}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a \cos \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = a \cos \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \cos \left (\lambda x \right )^{m}
\] |
[_Riccati] |
✓ |
|
|
\[
{}\left (a \cos \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )-a \,\lambda ^{2} \cos \left (\lambda x \right ) = 0
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a \lambda +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+\lambda ^{2}+3 a \lambda +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a y^{2}+b \tan \left (x \right ) y+c
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a y^{2}+2 a b \tan \left (x \right ) y+b \left (a b -1\right ) \tan \left (x \right )^{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a \tan \left (\beta x \right ) y+a b \tan \left (\beta x \right )-b^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a x \tan \left (b x \right )^{m} y+a \tan \left (b x \right )^{m}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+a \,x^{k +1} \tan \left (x \right )^{m} y-a \tan \left (x \right )^{m}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a \tan \left (\lambda x \right )^{n} y^{2}-a \,b^{2} \tan \left (\lambda x \right )^{n +2}+b \lambda \tan \left (\lambda x \right )^{2}+b \lambda
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a \tan \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = a \tan \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \tan \left (\lambda x \right )^{m}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a \lambda +a \left (\lambda -a \right ) \cot \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+\lambda ^{2}+3 a \lambda +a \left (\lambda -a \right ) \cot \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}-2 a b \cot \left (a x \right ) y+b^{2}-a^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a \cot \left (\beta x \right ) y+a b \cot \left (\beta x \right )-b^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a x \cot \left (b x \right )^{m} y+a \cot \left (b x \right )^{m}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+a \,x^{k +1} \cot \left (x \right )^{m} y-a \cot \left (x \right )^{m}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = a \cot \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \cot \left (\lambda x \right )^{m}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a \sin \left (\lambda x \right ) y^{2}+b \sin \left (\lambda x \right ) \cos \left (\lambda x \right )^{n}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a \cos \left (\lambda x \right ) y^{2}+b \cos \left (\lambda x \right ) \sin \left (\lambda x \right )^{n}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \lambda \sin \left (\lambda x \right ) y^{2}+a \,x^{n} \cos \left (\lambda x \right ) y-a \,x^{n}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}-y \tan \left (x \right )+a \left (1-a \right ) \cot \left (x \right )^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}-m y \tan \left (x \right )+b^{2} \cos \left (x \right )^{2 m}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+m y \cot \left (x \right )+b^{2} \sin \left (x \right )^{2 m}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a \lambda +b \lambda +2 a b +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2}+b \left (\lambda -b \right ) \cot \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}-\frac {\lambda ^{2}}{2}-\frac {3 \lambda ^{2} \tan \left (\lambda x \right )^{2}}{4}+a \cos \left (\lambda x \right )^{2} \sin \left (\lambda x \right )^{n}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \lambda \sin \left (\lambda x \right ) y^{2}+a \sin \left (\lambda x \right ) y-a \tan \left (\lambda x \right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+\lambda \arcsin \left (x \right )^{n} y-a^{2}+a \lambda \arcsin \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+\lambda x \arcsin \left (x \right )^{n} y+\lambda \arcsin \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+\lambda \arcsin \left (x \right )^{n} \left (x^{k +1} y-1\right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = \lambda \arcsin \left (x \right )^{n} y^{2}+k y+\lambda \,b^{2} x^{2 k} \arcsin \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+\lambda x \arccos \left (x \right )^{n} y+\lambda \arccos \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+\lambda \arccos \left (x \right )^{n} \left (x^{k +1} y-1\right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = \lambda \arccos \left (x \right )^{n} y^{2}+k y+\lambda \,b^{2} x^{2 k} \arccos \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+\lambda \arctan \left (x \right )^{n} y-a^{2}+a \lambda \arctan \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+\lambda x \arctan \left (x \right )^{n} y+\lambda \arctan \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+\lambda \arctan \left (x \right )^{n} \left (x^{k +1} y-1\right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = \lambda \arctan \left (x \right )^{n} y^{2}+k y+\lambda \,b^{2} x^{2 k} \arctan \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+\lambda \operatorname {arccot}\left (x \right )^{n} y-a^{2}+a \lambda \operatorname {arccot}\left (x \right )^{n}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+\lambda x \operatorname {arccot}\left (x \right )^{n} y+\lambda \operatorname {arccot}\left (x \right )^{n}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+\lambda \operatorname {arccot}\left (x \right )^{n} \left (x^{k +1} y-1\right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = \lambda \operatorname {arccot}\left (x \right )^{n} y^{2}+k y+\lambda \,b^{2} x^{2 k} \operatorname {arccot}\left (x \right )^{n}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+f \left (x \right ) y-a^{2}-a f \left (x \right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = f \left (x \right ) y^{2}-a y-a b -b^{2} f \left (x \right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+x f \left (x \right ) y+f \left (x \right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = f \left (x \right ) y^{2}-a \,x^{n} f \left (x \right ) y+a n \,x^{n -1}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = -\left (n +1\right ) x^{n} y^{2}+x^{n +1} f \left (x \right ) y-f \left (x \right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = f \left (x \right ) y^{2}+n y+a \,x^{2 n} f \left (x \right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = f \left (x \right ) y^{2}+g \left (x \right ) y-a^{2} f \left (x \right )-a g \left (x \right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{\lambda x} y^{2}+a \,{\mathrm e}^{\lambda x} f \left (x \right ) y+\lambda f \left (x \right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = f \left (x \right ) y^{2}-a \,{\mathrm e}^{\lambda x} f \left (x \right ) y+a \lambda \,{\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = f \left (x \right ) y^{2}+\lambda y+a^{2} {\mathrm e}^{2 \lambda x} f \left (x \right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = f \left (x \right ) y^{2}-f \left (x \right ) \left (a \,{\mathrm e}^{\lambda x}+b \right ) y+a \lambda \,{\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = {\mathrm e}^{\lambda x} f \left (x \right ) y^{2}+\left (a f \left (x \right )-\lambda \right ) y+b \,{\mathrm e}^{-\lambda x} f \left (x \right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = -a \ln \left (x \right ) y^{2}+a f \left (x \right ) \left (x \ln \left (x \right )-x \right ) y-f \left (x \right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \lambda \sin \left (\lambda x \right ) y^{2}+f \left (x \right ) \cos \left (\lambda x \right ) y-f \left (x \right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}-f \left (x \right )^{2}+f^{\prime }\left (x \right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = f \left (x \right ) y^{2}-f \left (x \right ) g \left (x \right ) y+g^{\prime }\left (x \right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = -f^{\prime }\left (x \right ) y^{2}+f \left (x \right ) g \left (x \right ) y-g \left (x \right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = g \left (x \right ) \left (y-f \left (x \right )\right )^{2}+f^{\prime }\left (x \right )
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = f^{\prime }\left (x \right ) y^{2}+a \,{\mathrm e}^{\lambda x} f \left (x \right ) y+a \,{\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = f \left (x \right ) y^{2}+g^{\prime }\left (x \right ) y+a f \left (x \right ) {\mathrm e}^{2 g \left (x \right )}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}-\frac {f^{\prime \prime }\left (x \right )}{f \left (x \right )}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y+2 x y^{2}-x^{2} y^{3}+2 x^{2} y y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
|
|
\[
{}-y+y^{\prime } x = x^{2}+y^{2}
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x -a y+b y^{2} = c \,x^{2 a}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime }+2 x y = x^{2}+y^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}R^{\prime } = \left (1+t \right ) \left (1+R^{2}\right )
\] |
[_separable] |
✓ |
|
|
\[
{}x^{\prime } = \left (4 t -x\right )^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}T^{\prime } = 2 a t \left (T^{2}-a^{2}\right )
\] |
[_separable] |
✓ |
|
|
\[
{}x^{\prime } = \left (t +x\right )^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}2 r \left (s^{2}+1\right )+\left (r^{4}+1\right ) s^{\prime } = 0
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = \left (1-x \right ) y^{2}+\left (2 x -1\right ) y-x
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = -y^{2}+x y+1
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = -8 x y^{2}+4 x \left (1+4 x \right ) y-8 x^{3}-4 x^{2}+1
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}2 x^{2}+x y+y^{2}+2 x^{2} y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \left (1+y^{2}\right ) \tan \left (x \right )
\] |
[_separable] |
✓ |
|
|
\[
{}x y+y^{2}+x^{2}-x^{2} y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = 1+y^{2}
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = \frac {1+y^{2}}{x^{2}+1}
\] |
[_separable] |
✓ |
|
|
\[
{}1+s^{2}-\sqrt {t}\, s^{\prime } = 0
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = \left (1+y^{2}\right ) t
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = \left (y+\frac {1}{2}\right ) \left (y+t \right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \left (y-2\right ) \left (y+1-\cos \left (t \right )\right )
\] |
[_Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime }+x y^{2} = x
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime }+\left (8-x \right ) y-y^{2} = -8 x
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = \left (x -y\right )^{2}
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \frac {1+y^{2}}{x^{2}+1}
\] |
[_separable] |
✓ |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime } = 1+y^{2}
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime }-3 x^{2} y^{2} = -3 x^{2}
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime }-3 x^{2} y^{2} = 3 x^{2}
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime }-x y^{2} = \sqrt {x}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = 1+\left (x y+3 y\right )^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = 1+\left (y-x \right )^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \left (x -y+3\right )^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = x \left (1+\frac {2 y}{x^{2}}+\frac {y^{2}}{x^{4}}\right )
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = y^{2}-2 x y+x^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = x y^{2}+3 y^{2}+x +3
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = 4 t^{2}-t y^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime } = t^{2} y^{2}+y^{2}-t^{2}-1
\] |
[_separable] |
✓ |
|
|
\[
{}4 \left (x -1\right )^{2} y^{\prime }-3 \left (3+y\right )^{2} = 0
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = \left (x +y-4\right )^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}1+y^{2}+\left (x^{2}+1\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
|
|
\[
{}1+y^{2} = y^{\prime } x
\] |
[_separable] |
✓ |
|
|
\[
{}a^{2}+y^{2}+2 x \sqrt {a x -x^{2}}\, y^{\prime } = 0
\] |
[_separable] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = y^{2}-x y+x^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}2 x^{2} y^{\prime } = x^{2}+y^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}{\mathrm e}^{-x} y^{\prime }+y^{2}-2 y \,{\mathrm e}^{x} = 1-{\mathrm e}^{2 x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime }+y^{2}-2 \sin \left (x \right ) y+\sin \left (x \right )^{2}-\cos \left (x \right ) = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x -y^{2}+\left (2 x +1\right ) y = x^{2}+2 x
\] |
[[_1st_order, _with_linear_symmetries], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = x^{2} y^{2}+x y+1
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \left (x -y\right )^{2}+1
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \left (1+y^{2}\right ) \tan \left (2 x \right )
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = 2 \left (x +1\right ) \left (1+y^{2}\right )
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime }+3 t y = 4-4 t^{2}+y^{2}
\] |
[_Riccati] |
✓ |
|
|
\[
{}y^{\prime }+y^{2}+\frac {y}{x}-\frac {4}{x^{2}} = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x -3 y+y^{2} = 4 x^{2}-4 x
\] |
[_rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = y+x^{2}+y^{2}
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
|
|
\[
{}1+y^{2}+\left (x^{2}+1\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
|
|
\[
{}y^{\prime } = \left (x +y\right )^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}1 = \frac {y}{1-x^{2} y^{2}}+\frac {x y^{\prime }}{1-x^{2} y^{2}}
\] |
[_exact, _rational, _Riccati] |
✓ |
|
|
\[
{}\frac {y^{\prime } x +y}{1-x^{2} y^{2}}+x = 0
\] |
[_exact, _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = x^{5}+x^{3} y^{2}+y
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } x = y+x^{2}+9 y^{2}
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = 1+\frac {y}{x}-\frac {y^{2}}{x^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}x^{2} y^{\prime } = x^{2}+x y+y^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \left (x +y\right )^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = x \left (a y^{2}+b \right )
\] |
[_separable] |
✓ |
|
|
\[
{}n^{\prime } = \left (n^{2}+1\right ) x
\] |
[_separable] |
✓ |
|
|
\[
{}3 x^{2} y^{\prime }+2 x^{2}-3 y^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
|
|
\[
{}y^{\prime }+2 x y = x^{2}+y^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|
|
\[
{}y^{\prime } = \left (4 x +y+1\right )^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
|