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\[
{} y y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0
\]
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\[
{} y y^{\prime \prime }+\sqrt {{y^{\prime }}^{2}+a^{2} {y^{\prime \prime }}^{2}} = {y^{\prime }}^{2}
\]
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\[
{} a y^{\prime \prime } = y^{\prime }
\]
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\[
{} a^{2} y^{\prime \prime } y^{\prime } = x
\]
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\[
{} y^{\prime \prime } = 1+{y^{\prime }}^{2}
\]
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\[
{} a y^{\prime \prime } = \sqrt {1+{y^{\prime }}^{2}}
\]
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\[
{} y^{\prime \prime } = a^{2}+k^{2} {y^{\prime }}^{2}
\]
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\[
{} a^{2} {y^{\prime \prime }}^{2} = 1+{y^{\prime }}^{2}
\]
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\[
{} y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0
\]
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\[
{} y^{\prime } = x y^{\prime \prime }+\sqrt {1+{y^{\prime }}^{2}}
\]
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\[
{} x y y^{\prime \prime }+x {y^{\prime }}^{2} = 3 y y^{\prime }
\]
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\[
{} 2 x^{2} y y^{\prime \prime }+y^{2} = x^{2} {y^{\prime }}^{2}
\]
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\[
{} x^{2} y^{\prime \prime } = \sqrt {m \,x^{2} {y^{\prime }}^{3}+n y^{2}}
\]
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\[
{} x^{4} y^{\prime \prime } = \left (x^{3}+2 x y\right ) y^{\prime }-4 y^{2}
\]
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\[
{} x^{4} y^{\prime \prime }-x^{3} y^{\prime } = x^{2} {y^{\prime }}^{2}-4 y^{2}
\]
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\[
{} x^{2} y^{\prime \prime }+4 y^{2}-6 y = x^{4} {y^{\prime }}^{2}
\]
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\[
{} y^{\prime \prime } = {\mathrm e}^{y}
\]
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\[
{} y^{\prime \prime }+a^{2} y = 0
\]
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\[
{} y^{\prime \prime } = \frac {1}{\sqrt {a y}}
\]
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\[
{} \left (x^{2}+1\right ) y^{\prime \prime }+3 x y^{\prime }+y = 0
\]
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\[
{} -a y^{\prime \prime } = \left (1+{y^{\prime }}^{2}\right )^{{3}/{2}}
\]
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\[
{} \sin \left (y\right )^{3} y^{\prime \prime } = \cos \left (y\right )
\]
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\[
{} {\mathrm e}^{x} \left (x y^{\prime \prime }-y^{\prime }\right ) = x^{3}
\]
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\[
{} \left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime } = 2
\]
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\[
{} y y^{\prime \prime }+\sqrt {{y^{\prime }}^{2}+a^{2} {y^{\prime \prime }}^{2}} = {y^{\prime }}^{2}
\]
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\[
{} y^{\prime \prime }-x^{2} y^{\prime }+x y = x
\]
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\[
{} x y^{\prime \prime }-\left (x +3\right ) y^{\prime }+3 y = 0
\]
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\[
{} x y^{\prime \prime }+\left (1-x \right ) y^{\prime } = y+{\mathrm e}^{x}
\]
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\[
{} \left (1+x \right ) y^{\prime \prime }-2 \left (x +3\right ) y^{\prime }+\left (x +5\right ) y = {\mathrm e}^{x}
\]
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\[
{} \left (3-x \right ) y^{\prime \prime }-\left (9-4 x \right ) y^{\prime }+\left (6-3 x \right ) y = 0
\]
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\[
{} y^{\prime \prime }+x y^{\prime }-y = X
\]
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\[
{} x^{2} y^{\prime \prime }-\left (x^{2}+2 x \right ) y^{\prime }+\left (x +2\right ) y = x^{3} {\mathrm e}^{x}
\]
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\[
{} y^{\prime \prime }-a x y^{\prime }+a^{2} \left (x -1\right ) y = 0
\]
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\[
{} \left (2 x^{3}-a \right ) y^{\prime \prime }-6 x^{2} y^{\prime }+6 x y = 0
\]
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\[
{} y^{\prime \prime }+4 x y^{\prime }+4 x^{2} y = 0
\]
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\[
{} y^{\prime \prime }+\frac {2 y^{\prime }}{x}+n^{2} y = 0
\]
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\[
{} y^{\prime \prime }+\frac {2 y^{\prime }}{x} = n^{2} y
\]
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\[
{} y^{\prime \prime }-2 b x y^{\prime }+b^{2} x^{2} y = 0
\]
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\[
{} y^{\prime \prime }-2 b x y^{\prime }+b^{2} x^{2} y = x
\]
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\[
{} 4 x^{2} y^{\prime \prime }+4 x^{5} y^{\prime }+\left (x^{3}+6 x^{2}+4\right ) y = 0
\]
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\[
{} x^{2} y^{\prime \prime }+\left (-4 x^{2}+x \right ) y^{\prime }+\left (4 x^{2}-2 x +1\right ) y = 0
\]
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\[
{} y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime }+5 y = \sec \left (x \right ) {\mathrm e}^{x}
\]
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\[
{} y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime }-\left (a^{2}+1\right ) y = 0
\]
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\[
{} y^{\prime \prime }-\frac {2 y^{\prime }}{x}+\left (n^{2}+\frac {2}{x^{2}}\right ) y = 0
\]
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\[
{} y^{\prime \prime }+2 n \cot \left (n x \right ) y^{\prime }+\left (m^{2}-n^{2}\right ) y = 0
\]
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\[
{} y^{\prime \prime }-\frac {y^{\prime }}{\sqrt {x}}+\frac {y \left (-8+\sqrt {x}+x \right )}{4 x^{2}} = 0
\]
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\[
{} x^{2} y^{\prime \prime }-2 n x y^{\prime }+\left (a^{2} x^{2}+n^{2}+n \right ) y = 0
\]
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\[
{} y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-3\right ) y = {\mathrm e}^{x^{2}}
\]
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\[
{} y^{\prime \prime }+\tan \left (x \right ) y^{\prime }+\cos \left (x \right )^{2} y = 0
\]
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\[
{} y^{\prime \prime }+\frac {2 y^{\prime }}{x}+\frac {a^{2} y}{x^{4}} = 0
\]
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\[
{} \left (x^{3}-x \right ) y^{\prime \prime }+y^{\prime }+n^{2} x^{3} y = 0
\]
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\[
{} \left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+m^{2} y = 0
\]
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\[
{} y^{\prime \prime }-\cot \left (x \right ) y^{\prime }-\sin \left (x \right )^{2} y = 0
\]
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\[
{} y^{\prime \prime } \sin \left (x \right )^{2}+\sin \left (x \right ) \cos \left (x \right ) y^{\prime }+y = 0
\]
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\[
{} \left (x^{2}+1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}+1\right ) y^{\prime }+4 y = 0
\]
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\[
{} y^{\prime \prime }+\left (\tan \left (x \right )-1\right )^{2} y^{\prime }-n \left (n -1\right ) y \sec \left (x \right )^{4} = 0
\]
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\[
{} y^{\prime \prime }+\left (3 \sin \left (x \right )-\cot \left (x \right )\right ) y^{\prime }+2 \sin \left (x \right )^{2} y = 0
\]
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\[
{} 3 x^{2} y^{\prime \prime }+\left (-6 x^{2}+2\right ) y^{\prime }-4 y = 0
\]
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\[
{} x y^{\prime \prime }+\left (x -2\right ) y^{\prime }-2 y = x^{2}
\]
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\[
{} x^{2} y^{\prime \prime }+y^{\prime }-\left (x^{2}+1\right ) y = {\mathrm e}^{-x}
\]
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\[
{} \left (x +2\right ) y^{\prime \prime }-\left (5+2 x \right ) y^{\prime }+2 y = \left (1+x \right ) {\mathrm e}^{x}
\]
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\[
{} y^{\prime \prime }+y = x
\]
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\[
{} y^{\prime \prime }+y = \csc \left (x \right )
\]
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\[
{} y^{\prime \prime }+4 y = 4 \tan \left (2 x \right )
\]
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\[
{} \left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = \left (1-x \right )^{2}
\]
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\[
{} y^{\prime \prime }-y = \frac {2}{1+{\mathrm e}^{x}}
\]
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\[
{} \left (-x^{2}+1\right ) y^{\prime \prime }-4 x y^{\prime }-\left (x^{2}+1\right ) y = x
\]
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\[
{} x^{2} y^{\prime \prime }-2 x \left (1+x \right ) y^{\prime }+2 \left (1+x \right ) y = -4 x^{3}
\]
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\[
{} x y^{\prime }-y = \left (x -1\right ) \left (y^{\prime \prime }-x +1\right )
\]
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\[
{} x^{2} y y^{\prime \prime }+\left (x y^{\prime }-y\right )^{2} = 0
\]
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\[
{} x^{2} y^{\prime \prime }-2 x \left (1+x \right ) y^{\prime }+2 \left (1+x \right ) y = x^{3}
\]
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\[
{} \left (x^{2}+a \right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0
\]
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\[
{} y^{\prime \prime }-\frac {2 y^{\prime }}{x}+\left (n^{2}+\frac {2}{x^{2}}\right ) y = 0
\]
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\[
{} y^{\prime \prime }+2 x y^{\prime }+\left (x^{2}+1\right ) y = x^{3}+3 x
\]
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\[
{} \left (a^{2}-x^{2}\right ) y^{\prime \prime }-\frac {a^{2} y^{\prime }}{x}+\frac {x^{2} y}{a} = 0
\]
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\[
{} x^{4} y^{\prime \prime }+2 x^{3} y^{\prime }+n^{2} y = 0
\]
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\[
{} \left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+\frac {a^{2} y}{-x^{2}+1} = 0
\]
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\[
{} \left (2 x -1\right ) y^{\prime \prime }-2 y^{\prime }+\left (3-2 x \right ) y = 2 \,{\mathrm e}^{x}
\]
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\[
{} x^{2} y^{\prime \prime }+x y^{\prime }-y = 8 x^{3}
\]
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\[
{} y^{\prime \prime }+2 x y^{\prime }+\left (x^{2}+5\right ) y = x \,{\mathrm e}^{-\frac {x^{2}}{2}}
\]
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\[
{} x \left (-x^{2}+1\right )^{2} y^{\prime \prime }+\left (-x^{2}+1\right ) \left (3 x^{2}+1\right ) y^{\prime }+4 x \left (x^{2}+1\right ) y = 0
\]
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\[
{} y^{\prime \prime }+\left (1-\frac {2}{x^{2}}\right ) y = x^{2}
\]
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\[
{} \left (x^{3}-2 x^{2}\right ) y^{\prime \prime }+2 x^{2} y^{\prime }-12 \left (x -2\right ) y = 0
\]
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\[
{} x y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+\left (x +2\right ) y = \left (x -2\right ) {\mathrm e}^{2 x}
\]
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\[
{} x^{2} y^{\prime \prime }+x y^{\prime }-y = 0
\]
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\[
{} x^{2} y^{\prime \prime }+x y^{\prime }-9 y = 0
\]
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\[
{} \left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }-a^{2} y = 0
\]
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\[
{} x y^{\prime \prime } \left (x \cos \left (x \right )-2 \sin \left (x \right )\right )+\left (x^{2}+2\right ) y^{\prime } \sin \left (x \right )-2 y \left (x \sin \left (x \right )+\cos \left (x \right )\right ) = 0
\]
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\[
{} x y^{\prime \prime }-y^{\prime }+4 x^{3} y = x^{5}
\]
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\[
{} \left (x^{2}-1\right ) y^{\prime \prime }-\left (4 x^{2}-3 x -5\right ) y^{\prime }+\left (4 x^{2}-6 x -5\right ) y = {\mathrm e}^{2 x}
\]
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\[
{} \left (x^{2}-1\right ) y^{\prime \prime }+x y^{\prime } = m^{2} y
\]
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\[
{} y^{\prime \prime }+\left (1-\frac {1}{x}\right ) y^{\prime }+4 x^{2} y \,{\mathrm e}^{-2 x} = 4 \left (x^{3}+x^{2}\right ) {\mathrm e}^{-3 x}
\]
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\[
{} x y^{\prime \prime }+\left (x^{2}+1\right ) y^{\prime }+2 x y = 2 x
\]
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\[
{} \left (x +2\right ) y^{\prime \prime }-\left (5+2 x \right ) y^{\prime }+2 y = \left (1+x \right ) {\mathrm e}^{x}
\]
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\[
{} \left (-x^{2}+1\right ) y^{\prime \prime }+x y^{\prime }-y = x \left (-x^{2}+1\right )^{{3}/{2}}
\]
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\[
{} x^{2} y^{\prime \prime }-\left (x^{2}+2 x \right ) y^{\prime }+\left (x +2\right ) y = 0
\]
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\[
{} 2 y^{\prime \prime }+9 y^{\prime }-18 y = 0
\]
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\[
{} y^{\prime \prime }+n^{2} y = \sec \left (n x \right )
\]
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\[
{} y^{\prime \prime }-4 y^{\prime }+y = a \cos \left (2 x \right )
\]
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\[
{} y^{\prime \prime }-2 y^{\prime }+y = x \sin \left (x \right )
\]
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