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

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

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

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

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

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

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

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

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

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

\[ {}y y^{\prime } = \frac {3 y}{\left (a x +b \right )^{\frac {1}{3}} x^{\frac {5}{3}}}+\frac {3}{\left (a x +b \right )^{\frac {2}{3}} x^{\frac {7}{3}}} \]

\[ {}3 y y^{\prime } = \frac {\left (-7 \lambda s \left (3 s +4 \lambda \right ) x +6 s -2 \lambda \right ) y}{x^{\frac {1}{3}}}+\frac {6 \lambda s x -6}{x^{\frac {2}{3}}}+2 \left (\lambda s \left (3 s +4 \lambda \right ) x +5 \lambda \right ) \left (-\lambda s \left (3 s +4 \lambda \right ) x +3 s +4 \lambda \right ) x^{\frac {1}{3}} \]

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

\[ {}y y^{\prime }-\frac {a \left (1+\frac {2 b}{x^{2}}\right ) y}{2} = \frac {a^{2} \left (3 x +\frac {4 b}{x}\right )}{16} \]

\[ {}y y^{\prime }+\frac {a \left (13 x -20\right ) y}{14 x^{\frac {9}{7}}} = -\frac {3 a^{2} \left (-1+x \right ) \left (x -8\right )}{14 x^{\frac {11}{17}}} \]

\[ {}y y^{\prime }+\frac {5 a \left (23 x -16\right ) y}{56 x^{\frac {9}{7}}} = -\frac {3 a^{2} \left (-1+x \right ) \left (25 x -32\right )}{56 x^{\frac {11}{17}}} \]

\[ {}y y^{\prime }+\frac {a \left (19 x +85\right ) y}{26 x^{\frac {18}{13}}} = -\frac {3 a^{2} \left (-1+x \right ) \left (x +25\right )}{26 x^{\frac {23}{13}}} \]

\[ {}y y^{\prime }+\frac {a \left (13 x -18\right ) y}{15 x^{\frac {7}{5}}} = -\frac {4 a^{2} \left (-1+x \right ) \left (x -6\right )}{15 x^{\frac {9}{5}}} \]

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

\[ {}y y^{\prime }+\frac {3 a \left (19 x -14\right ) x^{\frac {7}{5}} y}{35} = -\frac {4 a^{2} \left (-1+x \right ) \left (9 x -14\right ) x^{\frac {9}{5}}}{35} \]

\[ {}y y^{\prime }+\frac {3 a \left (3 x +7\right ) y}{10 x^{\frac {13}{10}}} = -\frac {a^{2} \left (-1+x \right ) \left (x +9\right )}{5 x^{\frac {8}{5}}} \]

\[ {}y y^{\prime }+\frac {a \left (7 x -12\right ) y}{10 x^{\frac {7}{5}}} = -\frac {a^{2} \left (-1+x \right ) \left (x -16\right )}{10 x^{\frac {9}{5}}} \]

\[ {}y y^{\prime }+\frac {3 a \left (13 x -8\right ) y}{20 x^{\frac {7}{5}}} = -\frac {a^{2} \left (-1+x \right ) \left (27 x -32\right )}{20 x^{\frac {9}{5}}} \]

\[ {}y y^{\prime }+\frac {3 a \left (3 x +11\right ) y}{14 x^{\frac {10}{7}}} = -\frac {a^{2} \left (-1+x \right ) \left (x -27\right )}{14 x^{\frac {13}{7}}} \]

\[ {}y y^{\prime }-\frac {a \left (1+x \right ) y}{2 x^{\frac {7}{4}}} = \frac {a^{2} \left (-1+x \right ) \left (3 x +5\right )}{4 x^{\frac {5}{2}}} \]

\[ {}y y^{\prime }-\frac {a \left (1+x \right ) y}{2 x^{\frac {7}{4}}} = \frac {a^{2} \left (-1+x \right ) \left (x +5\right )}{4 x^{\frac {5}{2}}} \]

\[ {}y y^{\prime }-\frac {a \left (4 x +3\right ) y}{14 x^{\frac {8}{7}}} = -\frac {a^{2} \left (-1+x \right ) \left (16 x +5\right )}{14 x^{\frac {9}{7}}} \]

\[ {}y y^{\prime }+\frac {a \left (13 x -3\right ) y}{6 x^{\frac {2}{3}}} = -\frac {a^{2} \left (-1+x \right ) \left (5 x -1\right )}{6 x^{\frac {1}{3}}} \]

\[ {}y y^{\prime }-\frac {a \left (8 x -1\right ) y}{28 x^{\frac {8}{7}}} = \frac {a^{2} \left (-1+x \right ) \left (32 x +3\right )}{28 x^{\frac {9}{7}}} \]

\[ {}y y^{\prime }-\frac {a \left (5 x -4\right ) y}{x^{4}} = \frac {a^{2} \left (-1+x \right ) \left (3 x -1\right )}{x^{7}} \]

\[ {}y y^{\prime }-\frac {2 a \left (3 x -10\right ) y}{5 x^{4}} = \frac {a^{2} \left (-1+x \right ) \left (8 x -5\right )}{5 x^{7}} \]

\[ {}y y^{\prime }+\frac {a \left (39 x -4\right ) y}{42 x^{\frac {9}{7}}} = -\frac {a^{2} \left (-1+x \right ) \left (9 x -1\right )}{42 x^{\frac {11}{7}}} \]

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

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

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

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

\[ {}y y^{\prime }+\frac {a \left (33 x +2\right ) y}{30 x^{\frac {6}{5}}} = -\frac {a^{2} \left (-1+x \right ) \left (9 x -4\right )}{30 x^{\frac {7}{5}}} \]

\[ {}y y^{\prime }-\frac {a \left (x -8\right ) y}{8 x^{\frac {5}{2}}} = -\frac {a^{2} \left (-1+x \right ) \left (3 x -4\right )}{8 x^{4}} \]

\[ {}y y^{\prime }+\frac {a \left (17 x +18\right ) y}{30 x^{\frac {22}{15}}} = -\frac {a^{2} \left (-1+x \right ) \left (4+x \right )}{30 x^{\frac {29}{15}}} \]

\[ {}y y^{\prime }-\frac {a \left (6 x -13\right ) y}{13 x^{\frac {5}{2}}} = -\frac {a^{2} \left (-1+x \right ) \left (x -13\right )}{26 x^{4}} \]

\[ {}y y^{\prime }+\frac {a \left (24 x +11\right ) x^{\frac {27}{20}} y}{30} = -\frac {a^{2} \left (-1+x \right ) \left (9 x +1\right )}{60 x^{\frac {17}{10}}} \]

\[ {}y y^{\prime }-\frac {2 a \left (2+3 x \right ) y}{5 x^{\frac {8}{5}}} = \frac {a^{2} \left (-1+x \right ) \left (8 x +1\right )}{5 x^{\frac {11}{5}}} \]

\[ {}y y^{\prime }-\frac {6 a \left (4 x +1\right ) y}{5 x^{\frac {7}{5}}} = \frac {a^{2} \left (-1+x \right ) \left (27 x +8\right )}{5 x^{\frac {9}{5}}} \]

\[ {}y y^{\prime }-\frac {a \left (4+x \right ) y}{5 x^{\frac {8}{5}}} = \frac {a^{2} \left (-1+x \right ) \left (3 x +7\right )}{5 x^{\frac {3}{5}}} \]

\[ {}y y^{\prime }-\frac {a \left (4+x \right ) y}{5 x^{\frac {8}{5}}} = \frac {a^{2} \left (-1+x \right ) \left (3 x +7\right )}{5 x^{\frac {11}{5}}} \]

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

\[ {}y y^{\prime }+\frac {a \left (x -6\right ) y}{5 x^{\frac {7}{5}}} = \frac {2 a^{2} \left (-1+x \right ) \left (4+x \right )}{5 x^{\frac {9}{5}}} \]

\[ {}y y^{\prime }+\frac {a \left (21 x +19\right ) y}{5 x^{\frac {7}{5}}} = -\frac {2 a^{2} \left (-1+x \right ) \left (9 x -4\right )}{5 x^{\frac {9}{5}}} \]

\[ {}y y^{\prime }-\frac {3 a y}{x^{\frac {7}{4}}} = \frac {a^{2} \left (-1+x \right ) \left (x -9\right )}{4 x^{\frac {5}{2}}} \]

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

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

\[ {}y y^{\prime }-\frac {a \left (\left (4 k -7\right ) x -4 k +5\right ) x^{-k} y}{2} = \frac {a^{2} \left (2 k -3\right ) \left (-1+x \right )^{2} x^{1-2 k}}{2} \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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