\[ {}y^{\prime } y-y = \frac {9 x}{32}+\frac {15 \sqrt {b^{2}+x^{2}}}{32}+\frac {3 b^{2}}{64 \sqrt {b^{2}+x^{2}}} \]

\[ {}y^{\prime } y-y = -\frac {3 x}{32}-\frac {3 \sqrt {a^{2}+x^{2}}}{32}+\frac {15 a^{2}}{64 \sqrt {a^{2}+x^{2}}} \]

\[ {}y^{\prime } y-y = A \,x^{2}-\frac {9}{625 A} \]

\[ {}y^{\prime } y-y = -\frac {6}{25} x -A \,x^{2} \]

\[ {}y^{\prime } y-y = \frac {6}{25} x -A \,x^{2} \]

\[ {}y^{\prime } y-y = 12 x +\frac {A}{x^{\frac {5}{2}}} \]

\[ {}y^{\prime } y-y = \frac {63 x}{4}+\frac {A}{x^{\frac {5}{3}}} \]

\[ {}y^{\prime } y-y = 2 x +2 A \left (10 \sqrt {x}+31 A +\frac {30 A^{2}}{\sqrt {x}}\right ) \]

\[ {}y^{\prime } y-y = 2 x +2 A \left (-10 \sqrt {x}+19 A +\frac {30 A^{2}}{\sqrt {x}}\right ) \]

\[ {}y^{\prime } y-y = -\frac {28 x}{121}+\frac {2 A \left (5 \sqrt {x}+106 A +\frac {65 A^{2}}{\sqrt {x}}\right )}{121} \]

\[ {}y^{\prime } y-y = -\frac {12 x}{49}+\frac {A \left (5 \sqrt {x}+262 A +\frac {65 A^{2}}{\sqrt {x}}\right )}{49} \]

\[ {}y^{\prime } y-y = -\frac {12 x}{49}+A \sqrt {x} \]

\[ {}y^{\prime } y-y = 6 x +\frac {A}{x^{4}} \]

\[ {}y^{\prime } y-y = 20 x +\frac {A}{\sqrt {x}} \]

\[ {}y^{\prime } y-y = \frac {15 x}{4}+\frac {A}{x^{7}} \]

\[ {}y^{\prime } y-y = -\frac {10 x}{49}+\frac {2 A \left (4 \sqrt {x}+61 A +\frac {12 A^{2}}{\sqrt {x}}\right )}{49} \]

\[ {}y^{\prime } y-y = -\frac {12 x}{49}+\frac {2 A \left (\sqrt {x}+166 A +\frac {55 A^{2}}{\sqrt {x}}\right )}{49} \]

\[ {}y^{\prime } y-y = -\frac {4 x}{25}+\frac {A \left (7 \sqrt {x}+49 A +\frac {6 A^{2}}{\sqrt {x}}\right )}{50} \]

\[ {}y^{\prime } y-y = \frac {15 x}{4}+\frac {6 A}{x^{\frac {1}{3}}}-\frac {3 A^{2}}{x^{\frac {5}{3}}} \]

\[ {}y^{\prime } y-y = -\frac {3 x}{16}+\frac {A}{x^{\frac {1}{3}}}+\frac {B}{x^{\frac {5}{3}}} \]

\[ {}y^{\prime } y-y = -\frac {5 x}{36}+\frac {A}{x^{\frac {3}{5}}}-\frac {B}{x^{\frac {7}{5}}} \]

\[ {}y^{\prime } y-y = \frac {k}{\sqrt {A \,x^{2}+B x +c}} \]

\[ {}y^{\prime } y-y = -\frac {12 x}{49}+3 A \left (\frac {1}{49}+B \right ) \sqrt {x}+3 A^{2} \left (\frac {4}{49}-\frac {5 B}{2}\right )+\frac {15 A^{3} \left (\frac {1}{49}-\frac {5 B}{4}\right )}{4 \sqrt {x}} \]

\[ {}y^{\prime } y-y = -\frac {6 x}{25}+\frac {4 B^{2} \left (\left (-A +2\right ) x^{\frac {1}{3}}-\frac {3 B \left (2 A +1\right )}{2}+\frac {B^{2} \left (1-3 A \right )}{x^{\frac {1}{3}}}-\frac {A \,B^{3}}{x^{\frac {2}{3}}}\right )}{75} \]

\[ {}y^{\prime } y-y = \frac {3 x}{4}-\frac {3 A \,x^{\frac {1}{3}}}{2}+\frac {3 A^{2}}{4 x^{\frac {1}{3}}}-\frac {27 A^{4}}{625 x^{\frac {5}{3}}} \]

\[ {}y^{\prime } y-y = -\frac {6 x}{25}+\frac {7 A \,x^{\frac {1}{3}}}{5}+\frac {31 A^{2}}{3 x^{\frac {1}{3}}}-\frac {100 A^{4}}{3 x^{\frac {5}{3}}} \]

\[ {}y^{\prime } y-y = -\frac {10 x}{49}+\frac {13 A^{2}}{5 x^{\frac {1}{5}}}-\frac {7 A^{3}}{20 x^{\frac {4}{5}}} \]

\[ {}y^{\prime } y-y = -\frac {33 x}{169}+\frac {286 A^{2}}{3 x^{\frac {5}{11}}}-\frac {770 A^{3}}{9 x^{\frac {13}{11}}} \]

\[ {}y^{\prime } y-y = -\frac {21 x}{100}+\frac {7 A^{2} \left (\frac {123}{x^{\frac {1}{7}}}+\frac {280 A}{x^{\frac {5}{7}}}-\frac {400 A^{2}}{x^{\frac {9}{7}}}\right )}{9} \]

\[ {}y^{\prime } y-y = a x +b \,x^{m} \]

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

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

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

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

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

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

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

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

\[ {}y^{\prime } y = \frac {y}{\sqrt {a x +b}}+1 \]

\[ {}y^{\prime } y = \frac {3 y}{\sqrt {a \,x^{\frac {3}{2}}+8 x}}+1 \]

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

\[ {}y^{\prime } y = a \,{\mathrm e}^{\lambda x} y+1 \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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