|
# |
ODE |
Mathematica |
Maple |
Sympy |
|
\[
{} x y^{\prime } = a y^{2}+\left (n +b \,x^{n}\right ) y+c \,x^{2 n}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} x y^{\prime } = x y^{2}+a y+b \,x^{n}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} x y^{\prime }+a_{3} x y^{2}+a_{2} y+a_{1} x +a_{0} = 0
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} x y^{\prime } = a \,x^{n} y^{2}+b y+c \,x^{-n}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} x y^{\prime } = a \,x^{n} y^{2}+m y-a \,b^{2} x^{n +2 m}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} x y^{\prime } = x^{2 n} y^{2}+\left (m -n \right ) y+x^{2 m}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} x y^{\prime } = a \,x^{n} y^{2}+b y+c \,x^{m}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} x y^{\prime } = a \,x^{2 n} y^{2}+\left (b \,x^{n}-n \right ) y+c
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} x y^{\prime } = a \,x^{2 n +m} y^{2}+\left (b \,x^{m +n}-n \right ) y+c \,x^{m}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} \left (a_{2} x +b_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (a_{1} x +b_{1} \right ) y+a_{0} x +b_{0} = 0
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} 2 y^{\prime } x^{2} = 2 y^{2}+x y-2 a^{2} x
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} 2 y^{\prime } x^{2} = 2 y^{2}+3 x y-2 a^{2} x
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } x^{2} = c \,x^{2} y^{2}+\left (a \,x^{2}+b x \right ) y+\alpha \,x^{2}+\beta x +\gamma
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } x^{2} = a \,x^{2} y^{2}+b x y+c \,x^{n}+s
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } x^{2} = a \,x^{2} y^{2}+b x y+c \,x^{2 n}+s \,x^{n}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } x^{2} = c \,x^{2} y^{2}+\left (a \,x^{n}+b \right ) x y+\alpha \,x^{2 n}+\beta \,x^{n}+\gamma
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } x^{2} = \left (\alpha \,x^{2 n}+\beta \,x^{n}+\gamma \right ) y^{2}+\left (a \,x^{n}+b \right ) x y+c \,x^{2}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} \left (x^{2}-1\right ) y^{\prime }+\lambda \left (1-2 x y+y^{2}\right ) = 0
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} \left (a \,x^{2}+b \right ) y^{\prime }+\alpha y^{2}+\beta x y+\frac {b \left (a +\beta \right )}{\alpha } = 0
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} \left (a \,x^{2}+b \right ) y^{\prime }+\alpha y^{2}+\beta x y+\gamma = 0
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} \left (a \,x^{2}+b x +c \right ) y^{\prime } = y^{2}+\left (2 \lambda x +b \right ) y+\lambda \left (\lambda -a \right ) x^{2}+\mu
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} \left (a \,x^{2}+b x +c \right ) y^{\prime } = y^{2}+\left (a x +\mu \right ) y-\lambda ^{2} x^{2}+\lambda \left (b -\mu \right ) x +\lambda c
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} \left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime } = y^{2}+\left (a_{1} x +b_{1} \right ) y-\lambda \left (\lambda +a_{1} -a_{2} \right ) x^{2}+\lambda \left (b_{2} -b_{1} \right ) x +\lambda c_{2}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} \left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime } = y^{2}+\left (a_{1} x +b_{1} \right ) y+a_{0} x^{2}+b_{0} x +c_{0}
\]
|
✗ |
✓ |
✗ |
|
|
\[
{} \left (c_{2} x^{2}+b_{2} x +a_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (b_{1} x +a_{1} \right ) y+a_{0} = 0
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} x^{3} y^{\prime } = x^{3} a y^{2}+\left (b \,x^{2}+c \right ) y+s x
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} x^{3} y^{\prime } = x^{3} a y^{2}+x \left (b x +c \right ) y+\alpha x +\beta
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} x \left (x^{2}+a \right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (b \,x^{2}+c \right ) y+s x = 0
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} x^{2} \left (x +a \right ) \left (y^{\prime }+\lambda y^{2}\right )+x \left (b x +c \right ) y+\alpha x +\beta = 0
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} \left (a \,x^{2}+b x +e \right ) \left (x y^{\prime }-y\right )-y^{2}+x^{2} = 0
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} x^{2} \left (x^{2}+a \right ) \left (y^{\prime }+\lambda y^{2}\right )+x \left (b \,x^{2}+c \right ) y+s = 0
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} a \left (x^{2}-1\right ) \left (y^{\prime }+\lambda y^{2}\right )+b x \left (x^{2}-1\right ) y+c \,x^{2}+d x +s = 0
\]
|
✗ |
✗ |
✗ |
|
|
\[
{} x^{n +1} y^{\prime } = a \,x^{2 n} y^{2}+b \,x^{n} y+c \,x^{m}+d
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} x \left (a \,x^{k}+b \right ) y^{\prime } = \alpha \,x^{n} y^{2}+\left (\beta -a n \,x^{k}\right ) y+\gamma \,x^{-n}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} x^{2} \left (a \,x^{n}-1\right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (p \,x^{n}+q \right ) x y+r \,x^{n}+s = 0
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} \left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime } = c y^{2}-b \,x^{m -1} y+a \,x^{n -2}
\]
|
✗ |
✗ |
✗ |
|
|
\[
{} \left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime } = a \,x^{n -2} y^{2}+b \,x^{m -1} y+c
\]
|
✗ |
✗ |
✗ |
|
|
\[
{} \left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime } = \alpha \,x^{k} y^{2}+\beta \,x^{s} y-\alpha \,\lambda ^{2} x^{k}+\beta \lambda \,x^{s}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} \left (a \,x^{n}+b \,x^{m}+c \right ) \left (x y^{\prime }-y\right )+s \,x^{k} \left (y^{2}-\lambda \,x^{2}\right ) = 0
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = a y^{2}+b \,{\mathrm e}^{\lambda x}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = y^{2}+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} {\mathrm e}^{2 \lambda x}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = \sigma y^{2}+a +b \,{\mathrm e}^{\lambda x}+c \,{\mathrm e}^{2 \lambda x}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = \sigma y^{2}+a y+b \,{\mathrm e}^{x}+c
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = y^{2}+b y+a \left (\lambda -b \right ) {\mathrm e}^{\lambda x}-a^{2} {\mathrm e}^{2 \lambda x}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = y^{2}+a \,{\mathrm e}^{\lambda x} y-a b \,{\mathrm e}^{\lambda x}-b^{2}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = y^{2}+a \,{\mathrm e}^{2 \lambda x} \left ({\mathrm e}^{\lambda x}+b \right )^{n}-\frac {\lambda ^{2}}{4}
\]
|
✗ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = y^{2}+a \,{\mathrm e}^{8 \lambda x}+b \,{\mathrm e}^{6 \lambda x}+c \,{\mathrm e}^{4 \lambda x}-\lambda ^{2}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = a \,{\mathrm e}^{k x} y^{2}+b \,{\mathrm e}^{s x}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = b \,{\mathrm e}^{x \mu } y^{2}+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} b \,{\mathrm e}^{\left (\mu +2 \lambda \right ) x}
\]
|
✓ |
✗ |
✗ |
|
|
\[
{} y^{\prime } = a \,{\mathrm e}^{x \mu } y^{2}+\lambda y-a \,b^{2} {\mathrm e}^{\left (\mu +2 \lambda \right ) x}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = {\mathrm e}^{\lambda x} y^{2}+a \,{\mathrm e}^{x \mu } y+a \lambda \,{\mathrm e}^{\left (\mu -\lambda \right ) x}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = -\lambda \,{\mathrm e}^{\lambda x} y^{2}+a \,{\mathrm e}^{x \mu } y-a \,{\mathrm e}^{\left (\mu -\lambda \right ) x}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = a \,{\mathrm e}^{x \mu } y^{2}+a b \,{\mathrm e}^{x \left (\lambda +\mu \right )} y-b \lambda \,{\mathrm e}^{\lambda x}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = a \,{\mathrm e}^{k x} y^{2}+b y+c \,{\mathrm e}^{s x}+d \,{\mathrm e}^{-k x}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = a \,{\mathrm e}^{\left (\mu +2 \lambda \right ) x} y^{2}+\left (b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-\lambda \right ) y+c \,{\mathrm e}^{x \mu }
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = a \,{\mathrm e}^{k x} y^{2}+b y+c \,{\mathrm e}^{k n x}+d \,{\mathrm e}^{k \left (2 n +1\right ) x}
\]
|
✓ |
✗ |
✗ |
|
|
\[
{} y^{\prime } = {\mathrm e}^{x \mu } \left (y-b \,{\mathrm e}^{\lambda x}\right )^{2}+b \lambda \,{\mathrm e}^{\lambda x}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} \left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{x \mu }+c \right ) y^{\prime } = y^{2}+k \,{\mathrm e}^{\nu x} y-m^{2}+k m \,{\mathrm e}^{\nu x}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} \left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{x \mu }+c \right ) \left (y^{\prime }-y^{2}\right )+a \,\lambda ^{2} {\mathrm e}^{\lambda x}+b \,\mu ^{2} {\mathrm e}^{x \mu } = 0
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = y^{2}+a x \,{\mathrm e}^{\lambda x} y+{\mathrm e}^{\lambda x} a
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = a \,{\mathrm e}^{\lambda x} y^{2}+b n \,x^{n -1}-a \,b^{2} {\mathrm e}^{\lambda x} x^{2 n}
\]
|
✗ |
✗ |
✗ |
|
|
\[
{} y^{\prime } = {\mathrm e}^{\lambda x} y^{2}+a \,x^{n} y+a \lambda \,x^{n} {\mathrm e}^{-\lambda x}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = -\lambda \,{\mathrm e}^{\lambda x} y^{2}+a \,x^{n} {\mathrm e}^{\lambda x} y-a \,x^{n}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = a \,{\mathrm e}^{\lambda x} y^{2}-a b \,x^{n} {\mathrm e}^{\lambda x} y+b n \,x^{n -1}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = a \,x^{n} y^{2}+b \lambda \,{\mathrm e}^{\lambda x}-a \,b^{2} x^{n} {\mathrm e}^{2 \lambda x}
\]
|
✗ |
✗ |
✗ |
|
|
\[
{} y^{\prime } = a \,x^{n} y^{2}+\lambda y-a \,b^{2} x^{n} {\mathrm e}^{2 \lambda x}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = a \,x^{n} y^{2}-a b \,x^{n} {\mathrm e}^{\lambda x} y+b \lambda \,{\mathrm e}^{\lambda x}
\]
|
✓ |
✗ |
✗ |
|
|
\[
{} y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+a \,x^{k +1} {\mathrm e}^{\lambda x} y-{\mathrm e}^{\lambda x} a
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = a \,x^{n} y^{2}-a \,x^{n} \left (b \,{\mathrm e}^{\lambda x}+c \right ) y+c \,x^{n}
\]
|
✗ |
✗ |
✗ |
|
|
\[
{} y^{\prime } = a \,x^{n} {\mathrm e}^{2 \lambda x} y^{2}+\left (b \,x^{n} {\mathrm e}^{\lambda x}-\lambda \right ) y+c \,x^{n}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = a \,{\mathrm e}^{\lambda x} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} x y^{\prime } = a \,{\mathrm e}^{\lambda x} y^{2}+k y+a \,b^{2} x^{2 k} {\mathrm e}^{\lambda x}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} x y^{\prime } = a \,x^{2 n} {\mathrm e}^{\lambda x} y^{2}+\left (b \,x^{n} {\mathrm e}^{\lambda x}-n \right ) y+c \,{\mathrm e}^{\lambda x}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = y^{2}+2 a \lambda x \,{\mathrm e}^{\lambda \,x^{2}}-a^{2} {\mathrm e}^{2 \lambda \,x^{2}}
\]
|
✗ |
✗ |
✗ |
|
|
\[
{} y^{\prime } = a \,{\mathrm e}^{-\lambda \,x^{2}} y^{2}+\lambda x y+a \,b^{2}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = a \,x^{n} y^{2}+\lambda x y+a \,b^{2} x^{n} {\mathrm e}^{\lambda \,x^{2}}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} x^{4} \left (y^{\prime }-y^{2}\right ) = a +b \,{\mathrm e}^{\frac {k}{x}}+c \,{\mathrm e}^{\frac {2 k}{x}}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = y^{2}-a^{2}+a \lambda \sinh \left (\lambda x \right )-a^{2} \sinh \left (\lambda x \right )^{2}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = y^{2}+a \sinh \left (\beta x \right ) y+a b \sinh \left (\beta x \right )-b^{2}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = y^{2}+a x \sinh \left (b x \right )^{m} y+a \sinh \left (b x \right )^{m}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = \lambda \sinh \left (\lambda x \right ) y^{2}-\lambda \sinh \left (\lambda x \right )^{3}
\]
|
✗ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = \left (a \sinh \left (\lambda x \right )^{2}-\lambda \right ) y^{2}-a \sinh \left (\lambda x \right )^{2}+\lambda -a
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} \left (a \sinh \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \sinh \left (x \mu \right ) y-d^{2}+c d \sinh \left (x \mu \right )
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} \left (a \sinh \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )+a \,\lambda ^{2} \sinh \left (\lambda x \right ) = 0
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = \alpha y^{2}+\beta +\gamma \cosh \left (x \right )
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = y^{2}+a \cosh \left (\beta x \right ) y+a b \cosh \left (\beta x \right )-b^{2}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = y^{2}+a x \cosh \left (b x \right )^{m} y+a \cosh \left (b x \right )^{m}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = \left (a \cosh \left (\lambda x \right )^{2}-\lambda \right ) y^{2}+a +\lambda -a \cosh \left (\lambda x \right )^{2}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} 2 y^{\prime } = \left (a -\lambda +a \cosh \left (\lambda x \right )\right ) y^{2}+a +\lambda -a \cosh \left (\lambda x \right )
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = y^{2}-\lambda ^{2}+a \cosh \left (\lambda x \right )^{n} \sinh \left (\lambda x \right )^{-n -4}
\]
|
✗ |
✗ |
✗ |
|
|
\[
{} y^{\prime } = a \sinh \left (\lambda x \right ) y^{2}+b \sinh \left (\lambda x \right ) \cosh \left (\lambda x \right )^{n}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = a \cosh \left (\lambda x \right ) y^{2}+b \cosh \left (\lambda x \right ) \sinh \left (\lambda x \right )^{n}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} \left (a \cosh \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \cosh \left (x \mu \right ) y-d^{2}+c d \cosh \left (x \mu \right )
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} \left (a \cosh \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )+a \,\lambda ^{2} \cosh \left (\lambda x \right ) = 0
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = y^{2}+a \lambda -a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = y^{2}+3 a \lambda -\lambda ^{2}-a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = y^{2}+a x \tanh \left (b x \right )^{m} y+a \tanh \left (b x \right )^{m}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} \left (a \tanh \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \tanh \left (x \mu \right ) y-d^{2}+c d \tanh \left (x \mu \right )
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = y^{2}+a \lambda -a \left (a +\lambda \right ) \coth \left (\lambda x \right )^{2}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime } = y^{2}-\lambda ^{2}+3 a \lambda -a \left (a +\lambda \right ) \coth \left (\lambda x \right )^{2}
\]
|
✓ |
✓ |
✗ |
|