\[ {}y y^{\prime }-\frac {a \left (\frac {\left (n +4\right ) x}{n +2}-2\right ) x^{-\frac {2 n +1}{n}} y}{n} = \frac {a^{2} \left (2 x^{2}+\left (n^{2}+n -4\right ) x -\left (n -1\right ) \left (n +2\right )\right ) x^{-\frac {3 n +2}{n}}}{n \left (n +2\right )} \]

\[ {}y y^{\prime }+\frac {a \left (\frac {\left (3 n +5\right ) x}{2}+\frac {n -1}{n +1}\right ) x^{-\frac {n +4}{n +3}} y}{n +3} = -\frac {a^{2} \left (\left (n +1\right ) x^{2}-\frac {\left (n^{2}+2 n +5\right ) x}{n +1}+\frac {4}{n +1}\right ) x^{-\frac {n +5}{n +3}}}{2 n +6} \]

\[ {}y y^{\prime }-a \left (\frac {n +2}{n}+b \,x^{n}\right ) y = -\frac {a^{2} x \left (\frac {n +1}{n}+b \,x^{n}\right )}{n} \]

\[ {}y y^{\prime } = \left (a \,{\mathrm e}^{x}+b \right ) y+c \,{\mathrm e}^{2 x}-a b \,{\mathrm e}^{x}-b^{2} \]

\[ {}y y^{\prime } = \left (a \left (2 \mu +\lambda \right ) {\mathrm e}^{\lambda x}+b \right ) {\mathrm e}^{x \mu } y+\left (-a^{2} \mu \,{\mathrm e}^{2 \lambda x}-a b \,{\mathrm e}^{\lambda x}+c \right ) {\mathrm e}^{2 x \mu } \]

\[ {}y y^{\prime } = \left ({\mathrm e}^{\lambda x} a +b \right ) y+c \left (a^{2} {\mathrm e}^{2 \lambda x}+a b \left (\lambda x +1\right ) {\mathrm e}^{\lambda x}+b^{2} \lambda x \right ) \]

\[ {}y y^{\prime } = {\mathrm e}^{\lambda x} \left (2 a \lambda x +a +b \right ) y-{\mathrm e}^{2 \lambda x} \left (a^{2} \lambda \,x^{2}+a b x +c \right ) \]

\[ {}y y^{\prime } = {\mathrm e}^{a x} \left (2 x^{2} a +b +2 x \right ) y+{\mathrm e}^{2 a x} \left (-a \,x^{4}-b \,x^{2}+c \right ) \]

\[ {}y y^{\prime }+a \left (2 b x +1\right ) {\mathrm e}^{b x} y = -a^{2} b \,x^{2} {\mathrm e}^{2 b x} \]

\[ {}y y^{\prime }-a \left (1+2 n +2 n \left (n +1\right ) x \right ) {\mathrm e}^{\left (n +1\right ) x} y = -a^{2} n \left (n +1\right ) \left (n x +1\right ) x \,{\mathrm e}^{2 \left (n +1\right ) x} \]

\[ {}y y^{\prime }+a \left (1+2 b \sqrt {x}\right ) {\mathrm e}^{2 b \sqrt {x}} y = -a^{2} b \,x^{\frac {3}{2}} {\mathrm e}^{4 b \sqrt {x}} \]

\[ {}y y^{\prime } = \left (a \cosh \left (x \right )+b \right ) y-a b \sinh \left (x \right )+c \]

\[ {}y y^{\prime } = \left (a \sinh \left (x \right )+b \right ) y-a b \cosh \left (x \right )+c \]

\[ {}y y^{\prime } = \left (2 \ln \left (x \right )+a +1\right ) y+x \left (-\ln \left (x \right )^{2}-a \ln \left (x \right )+b \right ) \]

\[ {}y y^{\prime } = \left (2 \ln \left (x \right )^{2}+2 \ln \left (x \right )+a \right ) y+x \left (-\ln \left (x \right )^{4}-a \ln \left (x \right )^{2}+b \right ) \]

\[ {}y y^{\prime } = a x \cos \left (\lambda \,x^{2}\right ) y+x \]

\[ {}y y^{\prime } = a x \sin \left (\lambda \,x^{2}\right ) y+x \]

\[ {}\left (y A +B x +a \right ) y^{\prime }+B y+k x +b = 0 \]

\[ {}\left (y+a x +b \right ) y^{\prime } = \alpha y+\beta x +\gamma \]

\[ {}\left (y+a k \,x^{2}+b x +c \right ) y^{\prime } = -a y^{2}+2 a k x y+m y+k \left (k +b -m \right ) x +s \]

\[ {}\left (y+A \,x^{n}+a \right ) y^{\prime }+n A \,x^{n -1} y+k \,x^{m}+b = 0 \]

\[ {}\left (y+a \,x^{n +1}+b \,x^{n}\right ) y^{\prime } = \left (a n \,x^{n}+c \,x^{n -1}\right ) y \]

\[ {}x y y^{\prime } = a y^{2}+b y+c \,x^{n}+s \]

\[ {}x y y^{\prime } = -n y^{2}+a \left (2 n +1\right ) x y+b y-a^{2} n \,x^{2}-a b x +c \]

\[ {}y^{\prime \prime }+a y = 0 \]

\[ {}y^{\prime \prime }-\left (a x +b \right ) y = 0 \]

\[ {}y^{\prime \prime }-\left (x^{2} a^{2}+a \right ) y = 0 \]

\[ {}y^{\prime \prime }-\left (x^{2} a +b \right ) y = 0 \]

\[ {}y^{\prime \prime }+a^{3} x \left (-a x +2\right ) y = 0 \]

\[ {}y^{\prime \prime }-\left (x^{2} a +b c x \right ) y = 0 \]

\[ {}y^{\prime \prime }-a \,x^{n} y = 0 \]

\[ {}y^{\prime \prime }-a \left (a \,x^{2 n}+n \,x^{n -1}\right ) y = 0 \]

\[ {}y^{\prime \prime }-a \,x^{n -2} \left (a \,x^{n}+n +1\right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (a \,x^{2 n}+b \,x^{n -1}\right ) y = 0 \]

\[ {}y^{\prime \prime }+a y^{\prime }+b y = 0 \]

\[ {}y^{\prime \prime }+a y^{\prime }+\left (b x +c \right ) y = 0 \]

\[ {}y^{\prime \prime }+a y^{\prime }-\left (b \,x^{2}+c \right ) y = 0 \]

\[ {}y^{\prime \prime }+a y^{\prime }+b \left (-b \,x^{2}+a x +1\right ) y = 0 \]

\[ {}y^{\prime \prime }+a y^{\prime }+b x \left (-b \,x^{3}+a x +2\right ) y = 0 \]

\[ {}y^{\prime \prime }+a y^{\prime }+b \left (-b \,x^{2 n}+a \,x^{n}+n \,x^{n -1}\right ) y = 0 \]

\[ {}y^{\prime \prime }+a y^{\prime }+b \left (-b \,x^{2 n}-a \,x^{n}+n \,x^{n -1}\right ) y = 0 \]

\[ {}y^{\prime \prime }+x y^{\prime }+\left (n -1\right ) y = 0 \]

\[ {}y^{\prime \prime }-2 x y^{\prime }+2 n y = 0 \]

\[ {}y^{\prime \prime }+a x y^{\prime }+b y = 0 \]

\[ {}y^{\prime \prime }+a x y^{\prime }+b x y = 0 \]

\[ {}y^{\prime \prime }+a x y^{\prime }+\left (b x +c \right ) y = 0 \]

\[ {}y^{\prime \prime }+2 a x y^{\prime }+\left (b \,x^{4}+x^{2} a^{2}+c x +a \right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (a x +b \right ) y^{\prime }-a y = 0 \]

\[ {}y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+a y = 0 \]

\[ {}y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (a x +b -c \right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (a x +2 b \right ) y^{\prime }+\left (a b x +b^{2}-a \right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (c x +d \right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{2}+b x +1\right ) y = 0 \]

\[ {}y^{\prime \prime }+2 \left (a x +b \right ) y^{\prime }+\left (x^{2} a^{2}+2 a b x +c \right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (-c \,x^{2 n}+a \,x^{n +1}+b \,x^{n}+n \,x^{n -1}\right ) y = 0 \]

\[ {}y^{\prime \prime }+a \left (-b^{2}+x^{2}\right ) y^{\prime }-a \left (x +b \right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (x^{2} a +b \right ) y^{\prime }+c \left (x^{2} a +b -c \right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (x^{2} a +2 b \right ) y^{\prime }+\left (a b \,x^{2}-a x +b^{2}\right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (2 x^{2}+a \right ) y^{\prime }+\left (x^{4}+x^{2} a +b +2 x \right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (x^{2} a +b x \right ) y^{\prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (a b \,x^{2}+b x +2 a \right ) y^{\prime }+a^{2} \left (b \,x^{2}+1\right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (x^{2} a +b x +c \right ) y^{\prime }+x \left (a b \,x^{2}+b c +2 a \right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (x^{2} a +b x +c \right ) y^{\prime }+\left (a b \,x^{3}+a c \,x^{2}+b \right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (a \,x^{3}+2 b \right ) y^{\prime }+\left (a b \,x^{3}-x^{2} a +b^{2}\right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (a \,x^{3}+b x \right ) y^{\prime }+2 \left (2 x^{2} a +b \right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (a b \,x^{3}+b \,x^{2}+2 a \right ) y^{\prime }+a^{2} \left (b \,x^{3}+1\right ) y = 0 \]

\[ {}y^{\prime \prime }+a \,x^{n} y^{\prime } = 0 \]

\[ {}y^{\prime \prime }+a \,x^{n} y^{\prime }+b \,x^{n -1} y = 0 \]

\[ {}y^{\prime \prime }+2 a \,x^{n} y^{\prime }+a \left (a \,x^{2 n}+n \,x^{n -1}\right ) y = 0 \]

\[ {}y^{\prime \prime }+a \,x^{n} y^{\prime }+\left (b \,x^{2 n}+c \,x^{n -1}\right ) y = 0 \]

\[ {}y^{\prime \prime }+a \,x^{n} y^{\prime }-b \left (a \,x^{m +n}+b \,x^{2 m}+m \,x^{m -1}\right ) y = 0 \]

\[ {}y^{\prime \prime }+2 a \,x^{n} y^{\prime }+\left (a^{2} x^{2 n}+b \,x^{2 m}+a n \,x^{n -1}+c \,x^{m -1}\right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+c \left (a \,x^{n}+b -c \right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (a \,x^{n}+2 b \right ) y^{\prime }+\left (a b \,x^{n}-a \,x^{n -1}+b^{2}\right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (a b \,x^{n}+b \,x^{n -1}+2 a \right ) y^{\prime }+a^{2} \left (b \,x^{n}+1\right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (a b \,x^{n}+2 b \,x^{n -1}-x \,a^{2}\right ) y^{\prime }+a \left (a b \,x^{n}+b \,x^{n -1}-x \,a^{2}\right ) y = 0 \]

\[ {}y^{\prime \prime }+x^{n} \left (x^{2} a +\left (a c +b \right ) x +b c \right ) y^{\prime }-x^{n} \left (a x +b \right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}\right ) y^{\prime }-\left (a \,x^{n -1}+b \,x^{m -1}\right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}\right ) y^{\prime }+\left (a n \,x^{n -1}+b m \,x^{m -1}\right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}\right ) y^{\prime }+\left (a \left (n +1\right ) x^{n -1}+b \left (m +1\right ) x^{m -1}\right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}\right ) y^{\prime }+c \left (a \,x^{n}+b \,x^{m}-c \right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}\right ) y^{\prime }+\left (a b \,x^{m +n}+b \left (m +1\right ) x^{m -1}-a \,x^{n -1}\right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime }+\left (a b \,x^{m +n}+b c \,x^{m}+a n \,x^{n -1}\right ) y = 0 \]

\[ {}x y^{\prime \prime }+\frac {y^{\prime }}{2}+a y = 0 \]

\[ {}x y^{\prime \prime }+a y^{\prime }+b y = 0 \]

\[ {}x y^{\prime \prime }+a y^{\prime }+b x y = 0 \]

\[ {}x y^{\prime \prime }+a y^{\prime }+\left (b x +c \right ) y = 0 \]

\[ {}x y^{\prime \prime }+n y^{\prime }+b \,x^{-2 n +1} y = 0 \]

\[ {}x y^{\prime \prime }+\left (1-3 n \right ) y^{\prime }-a^{2} n^{2} x^{2 n -1} y = 0 \]

\[ {}x y^{\prime \prime }+a y^{\prime }+b \,x^{n} y = 0 \]

\[ {}x y^{\prime \prime }+a y^{\prime }+b \,x^{n} \left (-b \,x^{n +1}+a +n \right ) y = 0 \]

\[ {}x y^{\prime \prime }+a x y^{\prime }+a y = 0 \]

\[ {}x y^{\prime \prime }+\left (b -x \right ) y^{\prime }-a y = 0 \]

\[ {}x y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (\left (a -c \right ) x +b \right ) y = 0 \]

\[ {}x y^{\prime \prime }+\left (2 a x +b \right ) y^{\prime }+a \left (a x +b \right ) y = 0 \]

\[ {}x y^{\prime \prime }+\left (x \left (a +b \right )+n +m \right ) y^{\prime }+\left (a b x +a n +b m \right ) y = 0 \]

\[ {}x y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (c x +d \right ) y = 0 \]

\[ {}x y^{\prime \prime }-\left (a x +1\right ) y^{\prime }-b \,x^{2} \left (b x +a \right ) y = 0 \]

\[ {}x y^{\prime \prime }-\left (2 a x +1\right ) y^{\prime }+\left (b \,x^{3}+x \,a^{2}+a \right ) y = 0 \]