\[ {}x^{4} y^{\prime \prime }+a \,x^{n} y^{\prime }-\left (a \,x^{n -1}+a b \,x^{n -2}+b^{2}\right ) y = 0 \]

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

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

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

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

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

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

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

\[ {}4 \left (x^{2}+1\right )^{2} y^{\prime \prime }+\left (x^{2} a +a -3\right ) y = 0 \]

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

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

\[ {}\left (-x^{2}+1\right )^{2} y^{\prime \prime }-2 x \left (-x^{2}+1\right ) y^{\prime }+\left (\nu \left (\nu +1\right ) \left (-x^{2}+1\right )-\mu ^{2}\right ) y = 0 \]

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

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

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

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

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

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

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

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

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

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

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

\[ {}x^{6} y^{\prime \prime }-x^{5} y^{\prime }+a y = 0 \]

\[ {}x^{6} y^{\prime \prime }+\left (3 x^{2}+a \right ) x^{3} y^{\prime }+b y = 0 \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}y^{\prime \prime }+a \,{\mathrm e}^{\lambda x} y = 0 \]

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

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

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

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

\[ {}y^{\prime \prime }+\left (a \,{\mathrm e}^{4 \lambda x}+b \,{\mathrm e}^{3 \lambda x}+c \,{\mathrm e}^{2 \lambda x}-\frac {\lambda ^{2}}{4}\right ) y = 0 \]

\[ {}y^{\prime \prime }+\left (a \,{\mathrm e}^{2 \lambda x} \left (b \,{\mathrm e}^{\lambda x}+c \right )^{n}-\frac {\lambda ^{2}}{4}\right ) y = 0 \]

\[ {}y^{\prime \prime }+a y^{\prime }+b \,{\mathrm e}^{2 a x} y = 0 \]

\[ {}y^{\prime \prime }-a y^{\prime }+b \,{\mathrm e}^{2 a x} y = 0 \]

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

\[ {}y^{\prime \prime }-y^{\prime }+\left (a \,{\mathrm e}^{3 \lambda x}+b \,{\mathrm e}^{2 \lambda x}+\frac {1}{4}-\frac {\lambda ^{2}}{4}\right ) y = 0 \]

\[ {}y^{\prime \prime }-y^{\prime }+\left (a \,{\mathrm e}^{2 \lambda x} \left (b \,{\mathrm e}^{\lambda x}+c \right )^{n}+\frac {1}{4}-\frac {\lambda ^{2}}{4}\right ) y = 0 \]

\[ {}y^{\prime \prime }+2 a \,{\mathrm e}^{\lambda x} y^{\prime }+a \,{\mathrm e}^{\lambda x} \left ({\mathrm e}^{\lambda x} a +\lambda \right ) y = 0 \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}y^{\prime \prime }+\left (2 \,{\mathrm e}^{\lambda x} a +b -\lambda \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+a b \,{\mathrm e}^{\lambda x}+c \,{\mathrm e}^{2 x \mu }+d \,{\mathrm e}^{x \mu }+k \right ) y = 0 \]

\[ {}y^{\prime \prime }+\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{x \mu }\right ) y^{\prime }+a \,{\mathrm e}^{\lambda x} \left (b \,{\mathrm e}^{x \mu }+\lambda \right ) y = 0 \]

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

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

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

\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = 0 \]

\[ {}y^{\prime \prime }-6 y^{\prime }+25 y = 0 \]

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

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

\[ {}4 x^{2} y^{\prime \prime }+4 x^{3} y^{\prime }+\left (x^{2}+1\right ) y = 0 \]

\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+4 y = 0 \]

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

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

\[ {}\left (x -3\right ) y^{\prime \prime }-\left (4 x -9\right ) y^{\prime }+\left (3 x -6\right ) y = 0 \]

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

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

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

\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (x^{2}+6\right ) y = 0 \]

\[ {}\left (2 x^{3}-1\right ) y^{\prime \prime }-6 x^{2} y^{\prime }+6 x y = 0 \]

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

\[ {}x^{4} y^{\prime \prime }+2 x^{3} \left (1+x \right ) y^{\prime }+n^{2} y = 0 \]

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

\[ {}2 y^{\prime \prime } = {\mathrm e}^{y} \]