| # | ODE | Mathematica | Maple | Sympy |
| \[
{} \left [x^{\prime }\left (t \right ) = \frac {x \left (t \right )^{2}}{2}-\frac {y \left (t \right )}{24}, y^{\prime }\left (t \right ) = 2 x \left (t \right ) y \left (t \right )-3 z \left (t \right ), z^{\prime }\left (t \right ) = 3 x \left (t \right ) z \left (t \right )-\frac {y \left (t \right )^{2}}{6}\right ]
\]
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| \[
{} [x^{\prime }\left (t \right ) = x \left (t \right ) \left (y \left (t \right )^{2}-z \left (t \right )^{2}\right ), y^{\prime }\left (t \right ) = y \left (t \right ) \left (z \left (t \right )^{2}-x \left (t \right )^{2}\right ), z^{\prime }\left (t \right ) = z \left (t \right ) \left (x \left (t \right )^{2}-y \left (t \right )^{2}\right )]
\]
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| \[
{} [x^{\prime }\left (t \right ) = x \left (t \right ) \left (y \left (t \right )^{2}-z \left (t \right )^{2}\right ), y^{\prime }\left (t \right ) = -y \left (t \right ) \left (z \left (t \right )^{2}+x \left (t \right )^{2}\right ), z^{\prime }\left (t \right ) = z \left (t \right ) \left (x \left (t \right )^{2}+y \left (t \right )^{2}\right )]
\]
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| \[
{} [x^{\prime }\left (t \right ) = -x \left (t \right ) y \left (t \right )^{2}+x \left (t \right )+y \left (t \right ), y^{\prime }\left (t \right ) = y \left (t \right ) x \left (t \right )^{2}-x \left (t \right )-y \left (t \right ), z^{\prime }\left (t \right ) = y \left (t \right )^{2}-x \left (t \right )^{2}]
\]
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| \[
{} y^{\prime } = y^{2}+k \left (a x +b \right )^{n} \left (c x +d \right )^{-n -4}
\]
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| \[
{} x^{2} y^{\prime } = x^{2} y^{2}+a \,x^{2 m} \left (b \,x^{m}+c \right )^{n}-\frac {n^{2}}{4}+\frac {1}{4}
\]
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| \[
{} a \,x^{2} \left (x -1\right )^{2} \left (y^{\prime }+\lambda y^{2}\right )+b \,x^{2}+c x +s = 0
\]
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| \[
{} y^{\prime } = -\left (n +1\right ) x^{n} y^{2}+a \,x^{n +m +1}-a \,x^{m}
\]
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| \[
{} y^{\prime } = a \,x^{n} y^{2}+b \,x^{m} y+c k \,x^{k -1}-b c \,x^{m +k}-a \,c^{2} x^{n +2 k}
\]
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| \[
{} \left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime } = y^{2}+\left (a_{1} x +b_{1} \right ) y+a_{0} x^{2}+b_{0} x +c_{0}
\]
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| \[
{} a \left (x^{2}-1\right ) \left (y^{\prime }+\lambda y^{2}\right )+b x \left (x^{2}-1\right ) y+c \,x^{2}+d x +s = 0
\]
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| \[
{} \left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime } = c y^{2}-b \,x^{m -1} y+a \,x^{n -2}
\]
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| \[
{} \left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime } = a \,x^{n -2} y^{2}+b \,x^{m -1} y+c
\]
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| \[
{} y^{\prime } = y^{2}+a \,{\mathrm e}^{2 \lambda x} \left ({\mathrm e}^{\lambda x}+b \right )^{n}-\frac {\lambda ^{2}}{4}
\]
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| \[
{} y^{\prime } = a \,{\mathrm e}^{\lambda x} y^{2}+b n \,x^{n -1}-a \,b^{2} {\mathrm e}^{\lambda x} x^{2 n}
\]
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| \[
{} y^{\prime } = a \,x^{n} y^{2}+b \lambda \,{\mathrm e}^{\lambda x}-a \,b^{2} x^{n} {\mathrm e}^{2 \lambda x}
\]
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| \[
{} y^{\prime } = a \,x^{n} y^{2}-a \,x^{n} \left (b \,{\mathrm e}^{\lambda x}+c \right ) y+c \,x^{n}
\]
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| \[
{} y^{\prime } = y^{2}+2 a \lambda x \,{\mathrm e}^{\lambda \,x^{2}}-a^{2} {\mathrm e}^{2 \lambda \,x^{2}}
\]
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| \[
{} y^{\prime } = \lambda \sinh \left (\lambda x \right ) y^{2}-\lambda \sinh \left (\lambda x \right )^{3}
\]
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| \[
{} y^{\prime } = y^{2}-\lambda ^{2}+a \cosh \left (\lambda x \right )^{n} \sinh \left (\lambda x \right )^{-n -4}
\]
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| \[
{} y^{\prime } = a \ln \left (x \right )^{n} y^{2}+b m \,x^{m -1}-a \,b^{2} x^{2 m} \ln \left (x \right )^{n}
\]
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| \[
{} x y^{\prime } = x y^{2}-a^{2} x \ln \left (\beta x \right )^{2}+a
\]
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| \[
{} x y^{\prime } = x y^{2}-a^{2} x \ln \left (\beta x \right )^{2 k}+a k \ln \left (\beta x \right )^{k -1}
\]
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| \[
{} x y^{\prime } = a \,x^{n} y^{2}+b -a \,b^{2} x^{n} \ln \left (x \right )^{2}
\]
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| \[
{} x^{2} y^{\prime } = x^{2} y^{2}+a \left (b \ln \left (x \right )+c \right )^{n}+\frac {1}{4}
\]
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| \[
{} y^{\prime } = a \,x^{n} y^{2}-a b \,x^{n +1} \ln \left (x \right ) y+b \ln \left (x \right )+b
\]
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| \[
{} y^{\prime } = y^{2}+\lambda ^{2}+c \sin \left (\lambda x +a \right )^{n} \sin \left (\lambda x +b \right )^{-n -4}
\]
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| \[
{} y^{\prime } = y^{2}+a \sin \left (b x \right )^{m} y+a \sin \left (b x \right )^{m}
\]
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| \[
{} y^{\prime } = \lambda \sin \left (\lambda x \right ) y^{2}+\lambda \sin \left (\lambda x \right )^{3}
\]
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| \[
{} 2 y^{\prime } = \left (\lambda +a -\sin \left (\lambda x \right ) a \right ) y^{2}+\lambda -a -\sin \left (\lambda x \right ) a
\]
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| \[
{} y^{\prime } = y^{2}+\lambda ^{2}+c \cos \left (\lambda x +a \right )^{n} \cos \left (\lambda x +b \right )^{-n -4}
\]
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| \[
{} y^{\prime } = y^{2}+a \cos \left (b x \right )^{m} y+a \cos \left (b x \right )^{m}
\]
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| \[
{} y^{\prime } = \lambda \cos \left (\lambda x \right ) y^{2}+\lambda \cos \left (\lambda x \right )^{3}
\]
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| \[
{} y^{\prime } = a \tan \left (\lambda x \right )^{n} y^{2}-a \,b^{2} \tan \left (\lambda x \right )^{n +2}+b \lambda \tan \left (\lambda x \right )^{2}+b \lambda
\]
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| \[
{} y^{\prime } = y^{2}+a \lambda +a \left (\lambda -a \right ) \cot \left (\lambda x \right )^{2}
\]
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| \[
{} y^{\prime } = y^{2}-2 a b \cot \left (a x \right ) y+b^{2}-a^{2}
\]
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| \[
{} y^{\prime } = -\left (1+k \right ) x^{k} y^{2}+a \,x^{1+k} \cot \left (x \right )^{m} y-a \cot \left (x \right )^{m}
\]
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| \[
{} y^{\prime } = y^{2}+\lambda ^{2}+c \sin \left (\lambda x \right )^{n} \cos \left (\lambda x \right )^{-n -4}
\]
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| \[
{} y^{\prime } = \lambda \sin \left (\lambda x \right ) y^{2}+a \,x^{n} \cos \left (\lambda x \right ) y-a \,x^{n}
\]
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| \[
{} y^{\prime } = y^{2}-2 \lambda ^{2} \tan \left (x \right )^{2}-2 \lambda ^{2} \cot \left (\lambda x \right )^{2}
\]
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| \[
{} y^{\prime } = y^{2}+a \lambda +b \lambda +2 a b +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2}+b \left (\lambda -b \right ) \cot \left (\lambda x \right )^{2}
\]
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| \[
{} y^{\prime } = y^{2}-\frac {\lambda ^{2}}{2}-\frac {3 \lambda ^{2} \tan \left (\lambda x \right )^{2}}{4}+a \cos \left (\lambda x \right )^{2} \sin \left (\lambda x \right )^{n}
\]
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| \[
{} y^{\prime } = \lambda \sin \left (\lambda x \right ) y^{2}+a \sin \left (\lambda x \right ) y-a \tan \left (\lambda x \right )
\]
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| \[
{} y^{\prime } = -\left (1+k \right ) x^{k} y^{2}+\lambda \arcsin \left (x \right )^{n} \left (x^{1+k} y-1\right )
\]
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| \[
{} y^{\prime } = \lambda \arcsin \left (x \right )^{n} y^{2}-b \lambda \,x^{m} \arcsin \left (x \right )^{n} y+b m \,x^{m -1}
\]
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| \[
{} y^{\prime } = \lambda \arcsin \left (x \right )^{n} y^{2}+\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \arcsin \left (x \right )^{n}
\]
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| \[
{} x y^{\prime } = \left (a \,x^{2 m} y^{2}+b \,x^{n} y+c \right ) \arcsin \left (x \right )^{m}-n y
\]
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| \[
{} y^{\prime } = -\left (1+k \right ) x^{k} y^{2}+\lambda \arccos \left (x \right )^{n} \left (x^{1+k} y-1\right )
\]
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| \[
{} y^{\prime } = \lambda \arccos \left (x \right )^{n} y^{2}-b \lambda \,x^{m} \arccos \left (x \right )^{n} y+b m \,x^{m -1}
\]
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| \[
{} y^{\prime } = \lambda \arccos \left (x \right )^{n} y^{2}+\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \arccos \left (x \right )^{n}
\]
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| \[
{} x y^{\prime } = \left (a \,x^{2 m} y^{2}+b \,x^{n} y+c \right ) \arccos \left (x \right )^{m}-n y
\]
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| \[
{} y^{\prime } = -\left (1+k \right ) x^{k} y^{2}+\lambda \arctan \left (x \right )^{n} \left (x^{1+k} y-1\right )
\]
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| \[
{} y^{\prime } = \lambda \arctan \left (x \right )^{n} y^{2}-b \lambda \,x^{m} \arctan \left (x \right )^{n} y+b m \,x^{m -1}
\]
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| \[
{} y^{\prime } = -\left (1+k \right ) x^{k} y^{2}+\lambda \operatorname {arccot}\left (x \right )^{n} \left (x^{1+k} y-1\right )
\]
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| \[
{} y^{\prime } = \lambda \operatorname {arccot}\left (x \right )^{n} y^{2}-b \lambda \,x^{m} \operatorname {arccot}\left (x \right )^{n} y+b m \,x^{m -1}
\]
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| \[
{} y^{\prime } = f \left (x \right ) y^{2}-a \,x^{n} f \left (x \right ) y+a n \,x^{n -1}
\]
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| \[
{} y^{\prime } = f \left (x \right ) y^{2}+a n \,x^{n -1}-a^{2} x^{2 n} f \left (x \right )
\]
|
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| \[
{} y^{\prime } = -\left (n +1\right ) x^{n} y^{2}+x^{n +1} f \left (x \right ) y-f \left (x \right )
\]
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✓ |
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| \[
{} y^{\prime } = f \left (x \right ) y^{2}+g \left (x \right ) y+a n \,x^{n -1}-a \,x^{n} g \left (x \right )-a^{2} x^{2 n} f \left (x \right )
\]
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| \[
{} y^{\prime } = f \left (x \right ) y^{2}-a \,x^{n} g \left (x \right ) y+a n \,x^{n -1}+a^{2} x^{2 n} \left (g \left (x \right )-f \left (x \right )\right )
\]
|
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| \[
{} y^{\prime } = f \left (x \right ) y^{2}+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} {\mathrm e}^{2 \lambda x} f \left (x \right )
\]
|
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| \[
{} y^{\prime } = f \left (x \right ) y^{2}-f \left (x \right ) \left ({\mathrm e}^{\lambda x} a +b \right ) y+a \lambda \,{\mathrm e}^{\lambda x}
\]
|
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| \[
{} y^{\prime } = f \left (x \right ) y^{2}+g \left (x \right ) y+a \lambda \,{\mathrm e}^{\lambda x}-a \,{\mathrm e}^{\lambda x} g \left (x \right )-a^{2} {\mathrm e}^{2 \lambda x} f \left (x \right )
\]
|
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| \[
{} y^{\prime } = f \left (x \right ) y^{2}-a \,{\mathrm e}^{\lambda x} g \left (x \right ) y+a \lambda \,{\mathrm e}^{\lambda x}+a^{2} {\mathrm e}^{2 \lambda x} \left (g \left (x \right )-f \left (x \right )\right )
\]
|
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| \[
{} y^{\prime } = f \left (x \right ) y^{2}+2 a \lambda x \,{\mathrm e}^{\lambda \,x^{2}}-a^{2} f \left (x \right ) {\mathrm e}^{2 \lambda \,x^{2}}
\]
|
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| \[
{} y^{\prime } = f \left (x \right ) y^{2}+\lambda x y+a f \left (x \right ) {\mathrm e}^{\lambda x}
\]
|
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| \[
{} y^{\prime } = f \left (x \right ) y^{2}-a \tanh \left (\lambda x \right )^{2} \left (f \left (x \right ) a +\lambda \right )+a \lambda
\]
|
✗ |
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| \[
{} y^{\prime } = f \left (x \right ) y^{2}-a \coth \left (\lambda x \right )^{2} \left (f \left (x \right ) a +\lambda \right )+a \lambda
\]
|
✗ |
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| \[
{} y^{\prime } = f \left (x \right ) y^{2}-a^{2} f \left (x \right )+a \lambda \sinh \left (\lambda x \right )-a^{2} f \left (x \right ) \sinh \left (\lambda x \right )^{2}
\]
|
✗ |
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| \[
{} x y^{\prime } = f \left (x \right ) y^{2}+a -a^{2} f \left (x \right ) \ln \left (x \right )^{2}
\]
|
✗ |
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| \[
{} y^{\prime } = f \left (x \right ) y^{2}-a x \ln \left (x \right ) f \left (x \right ) y+a \ln \left (x \right )+a
\]
|
✗ |
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| \[
{} y^{\prime } = -a \ln \left (x \right ) y^{2}+a f \left (x \right ) \left (x \ln \left (x \right )-x \right ) y-f \left (x \right )
\]
|
✗ |
✓ |
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| \[
{} y^{\prime } = \lambda \sin \left (\lambda x \right ) y^{2}+f \left (x \right ) \cos \left (\lambda x \right ) y-f \left (x \right )
\]
|
✗ |
✓ |
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|
| \[
{} y^{\prime } = f \left (x \right ) y^{2}-a^{2} f \left (x \right )+a \lambda \sin \left (\lambda x \right )+a^{2} f \left (x \right ) \sin \left (\lambda x \right )^{2}
\]
|
✗ |
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|
| \[
{} y^{\prime } = f \left (x \right ) y^{2}-a^{2} f \left (x \right )+a \lambda \cos \left (\lambda x \right )+a^{2} f \left (x \right ) \cos \left (\lambda x \right )^{2}
\]
|
✗ |
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|
| \[
{} y^{\prime } = f \left (x \right ) y^{2}-a \tan \left (\lambda x \right )^{2} \left (f \left (x \right ) a -\lambda \right )+a \lambda
\]
|
✗ |
✗ |
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|
| \[
{} y^{\prime } = f \left (x \right ) y^{2}-a \cot \left (\lambda x \right )^{2} \left (f \left (x \right ) a -\lambda \right )+a \lambda
\]
|
✗ |
✗ |
✗ |
|
| \[
{} y^{\prime } = y^{2}-f \left (x \right )^{2}+f^{\prime }\left (x \right )
\]
|
✗ |
✓ |
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|
| \[
{} y^{\prime } = f \left (x \right ) y^{2}-f \left (x \right ) g \left (x \right ) y+g^{\prime }\left (x \right )
\]
|
✗ |
✗ |
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| \[
{} y^{\prime } = -f^{\prime }\left (x \right ) y^{2}+f \left (x \right ) g \left (x \right ) y-g \left (x \right )
\]
|
✗ |
✓ |
✗ |
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| \[
{} f \left (x \right )^{2} y^{\prime }-f^{\prime }\left (x \right ) y^{2}+g \left (x \right ) \left (y-f \left (x \right )\right ) = 0
\]
|
✗ |
✗ |
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| \[
{} y^{\prime } = y^{2}+a^{2} f \left (a x +b \right )
\]
|
✗ |
✗ |
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| \[
{} y^{\prime } = y^{2}+\frac {f \left (\frac {1}{x}\right )}{x^{4}}
\]
|
✗ |
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|
| \[
{} y^{\prime } = y^{2}+\frac {f \left (\frac {a x +b}{c x +d}\right )}{\left (c x +d \right )^{4}}
\]
|
✗ |
✗ |
✗ |
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| \[
{} x^{2} y^{\prime } = x^{4} f \left (x \right ) y^{2}+1
\]
|
✗ |
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|
| \[
{} x^{2} y^{\prime } = x^{4} y^{2}+x^{2 n} f \left (a \,x^{n}+b \right )-\frac {n^{2}}{4}+\frac {1}{4}
\]
|
✗ |
✗ |
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| \[
{} y^{\prime } = f \left (x \right ) y^{2}+g \left (x \right ) y+h \left (x \right )
\]
|
✗ |
✗ |
✗ |
|
| \[
{} y^{\prime } = y^{2}+{\mathrm e}^{2 \lambda x} f \left ({\mathrm e}^{\lambda x}\right )-\frac {\lambda ^{2}}{4}
\]
|
✗ |
✗ |
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|
| \[
{} y^{\prime } = y^{2}-\frac {\lambda ^{2}}{4}+\frac {{\mathrm e}^{2 \lambda x} f \left (\frac {{\mathrm e}^{\lambda x} a +b}{c \,{\mathrm e}^{\lambda x}+d}\right )}{\left (c \,{\mathrm e}^{\lambda x}+d \right )^{4}}
\]
|
✗ |
✗ |
✗ |
|
| \[
{} y^{\prime } = y^{2}-\lambda ^{2}+\frac {f \left (\coth \left (\lambda x \right )\right )}{\sinh \left (\lambda x \right )^{4}}
\]
|
✗ |
✗ |
✗ |
|
| \[
{} y^{\prime } = y^{2}-\lambda ^{2}+\frac {f \left (\tanh \left (\lambda x \right )\right )}{\cosh \left (\lambda x \right )^{4}}
\]
|
✗ |
✗ |
✗ |
|
| \[
{} x^{2} y^{\prime } = x^{2} y^{2}+f \left (a \ln \left (x \right )+b \right )+\frac {1}{4}
\]
|
✗ |
✗ |
✗ |
|
| \[
{} y^{\prime } = y^{2}+\lambda ^{2}+\frac {f \left (\cot \left (\lambda x \right )\right )}{\sin \left (\lambda x \right )^{4}}
\]
|
✗ |
✗ |
✗ |
|
| \[
{} y^{\prime } = y^{2}+\lambda ^{2}+\frac {f \left (\tan \left (\lambda x \right )\right )}{\cos \left (\lambda x \right )^{4}}
\]
|
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|
| \[
{} y^{\prime } = y^{2}+\lambda ^{2}+\frac {f \left (\frac {\sin \left (\lambda x +a \right )}{\sin \left (\lambda x +b \right )}\right )}{\sin \left (\lambda x +b \right )^{4}}
\]
|
✗ |
✗ |
✗ |
|
| \[
{} y y^{\prime }-y = 2 A \left (\sqrt {x}+4 A +\frac {3 A^{2}}{\sqrt {x}}\right )
\]
|
✗ |
✓ |
✗ |
|
| \[
{} y y^{\prime }-y = A x +\frac {B}{x}-\frac {B^{2}}{x^{3}}
\]
|
✗ |
✓ |
✗ |
|
| \[
{} y y^{\prime }-y = A \,x^{k -1}-k B \,x^{k}+k \,B^{2} x^{2 k -1}
\]
|
✗ |
✗ |
✗ |
|
| \[
{} y y^{\prime }-y = A +B \,{\mathrm e}^{-\frac {2 x}{A}}
\]
|
✗ |
✓ |
✗ |
|
| \[
{} y y^{\prime }-y = A \left ({\mathrm e}^{\frac {2 x}{A}}-1\right )
\]
|
✗ |
✓ |
✗ |
|