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ODE |
Mathematica |
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\[
{}x^{n} y^{\prime \prime }+\left (a \,x^{n +m}+1\right ) y^{\prime }+a \,x^{m} \left (1+m \,x^{n -1}\right ) y = 0
\] |
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\[
{}\left (a \,x^{n}+b \right ) y^{\prime \prime }+\left (c \,x^{n}+d \right ) y^{\prime }+\lambda \left (\left (-a \lambda +c \right ) x^{n}+d -b \lambda \right ) y = 0
\] |
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\[
{}\left (a \,x^{n}+b x +c \right ) y^{\prime \prime } = a n \left (n -1\right ) x^{n -2} y
\] |
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\[
{}x^{2} \left (a^{2} x^{2 n}-1\right ) y^{\prime \prime }+x \left (a p \,x^{n}+q \right ) y^{\prime }+\left (a r \,x^{n}+s \right ) y = 0
\] |
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\[
{}\left (x^{n}+a \right )^{2} y^{\prime \prime }-b \,x^{n -2} \left (\left (b -1\right ) x^{n}+a \left (n -1\right )\right ) y = 0
\] |
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\[
{}\left (x^{n}+a \right )^{2} y^{\prime \prime }+b \,x^{m} \left (x^{n}+a \right ) y^{\prime }-x^{n -2} \left (b \,x^{m +1}+a n -a \right ) y = 0
\] |
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\[
{}\left (a \,x^{n}+b \right )^{2} y^{\prime \prime }+c \,x^{m} \left (a \,x^{n}+b \right ) y^{\prime }+\left (c \,x^{m}-a n \,x^{n -1}-1\right ) y = 0
\] |
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\[
{}\left (a \,x^{n +1}+b \,x^{n}+c \right )^{2} y^{\prime \prime }+\left (\alpha \,x^{n}+\beta \,x^{n -1}+\gamma \right ) y^{\prime }+\left (n \left (-a n -a +\alpha \right ) x^{n -1}+\left (n -1\right ) \left (-b n +\beta \right ) x^{n -2}\right ) y = 0
\] |
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\[
{}\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime \prime }+\left (\lambda -x \right ) y^{\prime }+y = 0
\] |
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\[
{}\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime \prime }+\left (\lambda ^{2}-x^{2}\right ) y^{\prime }+\left (x +\lambda \right ) y = 0
\] |
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\[
{}2 \left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime \prime }+a n \,x^{n -1} b m \,x^{m -1} y^{\prime }+y d = 0
\] |
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\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{2 \lambda x} \left (b \,{\mathrm e}^{\lambda x}+c \right )^{n}-\frac {\lambda ^{2}}{4}\right ) y = 0
\] |
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\[
{}y^{\prime \prime }-y^{\prime }+\left (a \,{\mathrm e}^{2 \lambda x} \left (b \,{\mathrm e}^{\lambda x}+c \right )^{n}+\frac {1}{4}-\frac {\lambda ^{2}}{4}\right ) y = 0
\] |
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\[
{}y^{\prime \prime }+a \,{\mathrm e}^{\lambda x} y^{\prime }-b \,{\mathrm e}^{x \mu } \left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{x \mu }+\mu \right ) y = 0
\] |
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\[
{}y^{\prime \prime }+\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{x \mu }\right ) y^{\prime }+a \,{\mathrm e}^{\lambda x} \left (b \,{\mathrm e}^{x \mu }+\lambda \right ) y = 0
\] |
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\[
{}y^{\prime \prime }+{\mathrm e}^{\lambda x} \left (a \,{\mathrm e}^{2 x \mu }+b \right ) y^{\prime }+\mu \left ({\mathrm e}^{\lambda x} \left (b -a \,{\mathrm e}^{2 x \mu }\right )-\mu \right ) y = 0
\] |
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\[
{}y^{\prime \prime }+\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{x \mu }+c \right ) y^{\prime }+\left (a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}+{\mathrm e}^{\lambda x} c a +b \mu \,{\mathrm e}^{x \mu }\right ) y = 0
\] |
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\[
{}y^{4} x^{3}+x^{2} y^{3}+x y^{2}+y+\left (y^{3} x^{4}-y^{2} x^{3}-x^{3} y+x \right ) y^{\prime } = 0
\] |
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\[
{}x +y^{\prime } y \left (2 {y^{\prime }}^{2}+3\right ) = 0
\] |
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\[
{}{y^{\prime }}^{3}-4 x y y^{\prime }+8 y^{2} = 0
\] |
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\[
{}{\mathrm e}^{2 y} {y^{\prime }}^{3}+\left ({\mathrm e}^{2 x}+{\mathrm e}^{3 x}\right ) y^{\prime }-{\mathrm e}^{3 x} = 0
\] |
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\[
{}\left (x -y^{\prime }-y\right )^{2} = x^{2} \left (2 x y-x^{2} y^{\prime }\right )
\] |
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\[
{}x^{2} y^{\prime \prime }-2 n x \left (1+x \right ) y^{\prime }+\left (a^{2} x^{2}+n^{2}+n \right ) y = 0
\] |
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\[
{}x^{4} y^{\prime \prime }+2 x^{3} \left (1+x \right ) y^{\prime }+n^{2} y = 0
\] |
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\[
{}x^{2} \left (-x^{3}+1\right ) y^{\prime \prime }-x^{3} y^{\prime }-2 y = 0
\] |
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\[
{}y^{\prime \prime }+2 \cot \left (x \right ) y^{\prime }+2 \tan \left (x \right ) {y^{\prime }}^{2} = 0
\] |
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\[
{}{x^{\prime }}^{2}+t x = \sqrt {t +1}
\] |
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\[
{}y^{\prime \prime }+y = 0
\] |
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\[
{}y^{\prime \prime }+y = 0
\] |
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\[
{}3 x^{2} y+2-\left (x^{3}+y\right ) y^{\prime } = 0
\] |
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\[
{}\left (2 t +1\right ) x^{\prime \prime }+t^{3} x^{\prime }+x = 0
\] |
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\[
{}\sin \left (t \right ) x^{\prime \prime }+\cos \left (t \right ) x^{\prime }+2 x = 0
\] |
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\[
{}f \left (t \right ) x^{\prime \prime }+g \left (t \right ) x = 0
\] |
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\[
{}[x^{\prime }\left (t \right ) = 4 x \left (t \right )-4 y \left (t \right )-x \left (t \right ) \left (x \left (t \right )^{2}+y \left (t \right )^{2}\right ), y^{\prime }\left (t \right ) = 4 x \left (t \right )+4 y \left (t \right )-y \left (t \right ) \left (x \left (t \right )^{2}+y \left (t \right )^{2}\right )]
\] |
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\[
{}\left [x^{\prime }\left (t \right ) = y \left (t \right )+\frac {x \left (t \right ) \left (1-x \left (t \right )^{2}-y \left (t \right )^{2}\right )}{\sqrt {x \left (t \right )^{2}+y \left (t \right )^{2}}}, y^{\prime }\left (t \right ) = -x \left (t \right )+\frac {y \left (t \right ) \left (1-x \left (t \right )^{2}-y \left (t \right )^{2}\right )}{\sqrt {x \left (t \right )^{2}+y \left (t \right )^{2}}}\right ]
\] |
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\[
{}x^{\prime \prime }+x^{4} x^{\prime }-x^{\prime }+x = 0
\] |
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\[
{}x^{\prime \prime }+x^{\prime }+{x^{\prime }}^{3}+x = 0
\] |
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\[
{}x^{\prime \prime }+\left (x^{4}+x^{2}\right ) x^{\prime }+x^{3}+x = 0
\] |
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\[
{}x^{\prime \prime }+\left (5 x^{4}-6 x^{2}\right ) x^{\prime }+x^{3} = 0
\] |
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\[
{}x^{\prime \prime }+\left (x^{2}+1\right ) x^{\prime }+x^{3} = 0
\] |
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\[
{}\left (\cos \left (t \right ) t -\sin \left (t \right )\right ) x^{\prime \prime }-x^{\prime } t \sin \left (t \right )-x \sin \left (t \right ) = 0
\] |
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\[
{}x = {y^{\prime }}^{3}-y^{\prime }+2
\] |
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\[
{}y = {y^{\prime }}^{4}-{y^{\prime }}^{3}-2
\] |
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\[
{}y^{\prime } = x y^{3}+x^{2}
\] |
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\[
{}y = x^{2}+2 x y^{\prime }+\frac {{y^{\prime }}^{2}}{2}
\] |
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\[
{}{y^{\prime \prime }}^{3}+y^{\prime \prime }+1 = x
\] |
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\[
{}y^{\prime } = \sin \left (x y\right )
\] |
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\[
{}y^{\prime } = \ln \left (x y\right )
\] |
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\[
{}y^{\prime \prime }+y y^{\prime \prime \prime \prime } = 1
\] |
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\[
{}y^{\prime \prime \prime }+x y^{\prime \prime }-y^{2} = \sin \left (x \right )
\] |
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\[
{}{y^{\prime }}^{2}+y {y^{\prime }}^{2} x = \ln \left (x \right )
\] |
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\[
{}\sin \left (y^{\prime \prime }\right )+y y^{\prime \prime \prime \prime } = 1
\] |
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\[
{}\sinh \left (x \right ) {y^{\prime }}^{2}+y^{\prime \prime } = x y
\] |
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\[
{}{y^{\prime \prime \prime }}^{2}+\sqrt {y} = \sin \left (x \right )
\] |
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\[
{}\left (x -3\right ) y^{\prime \prime }+\ln \left (x \right ) y = x^{2}
\] |
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\[
{}y^{\prime \prime }+\tan \left (x \right ) y^{\prime }+\cot \left (x \right ) y = 0
\] |
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\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+\left (x -1\right ) y^{\prime }+y = 0
\] |
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\[
{}x y^{\prime \prime }+2 x^{2} y^{\prime }+y \sin \left (x \right ) = \sinh \left (x \right )
\] |
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\[
{}\sin \left (x \right ) y^{\prime \prime }+x y^{\prime }+7 y = 1
\] |
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\[
{}y^{\prime \prime }+\frac {k x}{y^{4}} = 0
\] |
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\[
{}x y^{\prime \prime }+\sin \left (x \right ) y^{\prime }+y \cos \left (x \right ) = 0
\] |
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\[
{}\ln \left (x^{2}+1\right ) y^{\prime \prime }+\frac {4 x y^{\prime }}{x^{2}+1}+\frac {\left (-x^{2}+1\right ) y}{\left (x^{2}+1\right )^{2}} = 0
\] |
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\[
{}x y^{\prime \prime }+\left (6 x y^{2}+1\right ) y^{\prime }+2 y^{3}+1 = 0
\] |
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\[
{}y^{\prime \prime }+\frac {\left (x -1\right ) y^{\prime }}{x}+\frac {y}{x^{3}} = \frac {{\mathrm e}^{-\frac {1}{x}}}{x^{3}}
\] |
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\[
{}t^{2} y^{\prime \prime }-6 t y^{\prime }+\sin \left (2 t \right ) y = \ln \left (t \right )
\] |
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\[
{}y^{\prime \prime }+t y^{\prime }-y \ln \left (t \right ) = \cos \left (2 t \right )
\] |
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\[
{}y y^{\prime \prime }-{y^{\prime }}^{2}+{y^{\prime }}^{3} = 0
\] |
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\[
{}x^{\prime \prime }+x^{\prime }+x-x^{3} = 0
\] |
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\[
{}x^{\prime \prime }+x^{\prime }+x+x^{3} = 0
\] |
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\[
{}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0
\] |
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\[
{}y^{\prime } = y^{3}+x^{3}
\] |
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\[
{}y^{\prime } = \frac {1}{\sqrt {15-x^{2}-y^{2}}}
\] |
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\[
{}\sqrt {1-x}\, y^{\prime \prime }-4 y = \sin \left (x \right )
\] |
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\[
{}\left (x^{2}-4\right ) y^{\prime \prime }+\ln \left (x \right ) y = x \,{\mathrm e}^{x}
\] |
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\[
{}\left [y_{1}^{\prime }\left (x \right ) = \sin \left (x \right ) y_{1} \left (x \right )+\sqrt {x}\, y_{2} \left (x \right )+\ln \left (x \right ), y_{2}^{\prime }\left (x \right ) = \tan \left (x \right ) y_{1} \left (x \right )-{\mathrm e}^{x} y_{2} \left (x \right )+1\right ]
\] |
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\[
{}\left [y_{1}^{\prime }\left (x \right ) = \sin \left (x \right ) y_{1} \left (x \right )+\sqrt {x}\, y_{2} \left (x \right )+\ln \left (x \right ), y_{2}^{\prime }\left (x \right ) = \tan \left (x \right ) y_{1} \left (x \right )-{\mathrm e}^{x} y_{2} \left (x \right )+1\right ]
\] |
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\[
{}\left [y_{1}^{\prime }\left (x \right ) = {\mathrm e}^{-x} y_{1} \left (x \right )-\sqrt {1+x}\, y_{2} \left (x \right )+x^{2}, y_{2}^{\prime }\left (x \right ) = \frac {y_{1} \left (x \right )}{\left (x -2\right )^{2}}\right ]
\] |
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\[
{}\left [y_{1}^{\prime }\left (x \right ) = {\mathrm e}^{-x} y_{1} \left (x \right )-\sqrt {1+x}\, y_{2} \left (x \right )+x^{2}, y_{2}^{\prime }\left (x \right ) = \frac {y_{1} \left (x \right )}{\left (x -2\right )^{2}}\right ]
\] |
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\[
{}[y_{1}^{\prime }\left (x \right ) = 2 x y_{1} \left (x \right )-x^{2} y_{2} \left (x \right )+4 x, y_{2}^{\prime }\left (x \right ) = {\mathrm e}^{x} y_{1} \left (x \right )+3 \,{\mathrm e}^{-x} y_{2} \left (x \right )-\cos \left (3 x \right )]
\] |
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\[
{}S^{\prime } = S^{3}-2 S^{2}+S
\] |
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\[
{}S^{\prime } = S^{3}-2 S^{2}+S
\] |
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\[
{}S^{\prime } = S^{3}-2 S^{2}+S
\] |
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\[
{}y^{\prime } = {\mathrm e}^{\frac {2}{y}}
\] |
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\[
{}y^{\prime } = {\mathrm e}^{\frac {2}{y}}
\] |
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\[
{}y^{\prime } = 2 y^{3}+t^{2}
\] |
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\[
{}y^{\prime } = \cos \left (y\right )
\] |
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\[
{}w^{\prime } = w \cos \left (w\right )
\] |
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\[
{}w^{\prime } = w \cos \left (w\right )
\] |
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\[
{}w^{\prime } = w \cos \left (w\right )
\] |
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\[
{}y^{\prime } = \left (y-3\right ) \left (\sin \left (y\right ) \sin \left (t \right )+\cos \left (t \right )+1\right )
\] |
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\[
{}y^{\prime } = \left (y-1\right ) \left (y-2\right ) \left (y-{\mathrm e}^{\frac {t}{2}}\right )
\] |
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\[
{}y^{2} y^{\prime \prime } = 8 x^{2}
\] |
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\[
{}\sin \left (x +y\right )-y y^{\prime } = 0
\] |
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\[
{}y^{\prime } y^{2}+3 x^{2} y = \sin \left (x \right )
\] |
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\[
{}y^{\prime \prime } = 2 y y^{\prime }
\] |
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\[
{}y^{\prime \prime } = 2 y y^{\prime }
\] |
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\[
{}y^{\prime \prime }+x^{2} y^{\prime }+4 y = y^{3}
\] |
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\[
{}y y^{\prime \prime \prime }+6 y^{\prime \prime }+3 y^{\prime } = y
\] |
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\[
{}x^{3} y^{\prime \prime \prime }-4 y^{\prime \prime }+10 y^{\prime }-12 y = 0
\] |
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\[
{}[x^{\prime }\left (t \right ) = x \left (t \right ) y \left (t \right )-6 y \left (t \right ), y^{\prime }\left (t \right ) = x \left (t \right )-y \left (t \right )-5]
\] |
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