\[ {}y^{\prime } = 1+x^{2}+y^{2}+y^{4} x^{2} \]

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

\[ {}x^{3} y^{\prime } = 2 y \]

\[ {}y^{\prime } = \frac {-{\mathrm e}^{-x}+x}{{\mathrm e}^{y}+x} \]

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

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

\[ {}u^{\prime \prime }+u^{\prime }+\frac {u^{3}}{5} = \cos \left (t \right ) \]

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

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

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

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

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

\[ {}y^{\prime } = \sqrt {x^{2}+y^{2}} \]

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

\[ {}2 x^{2}+8 x y+y^{2}+\left (2 x^{2}+\frac {x y^{3}}{3}\right ) y^{\prime } = 0 \]

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

\[ {}3 x^{2} \cos \left (x \right ) y-x^{3} y^{2} \sin \left (x \right )+4 x +\left (8 y-x^{4} \sin \left (x \right ) y\right ) y^{\prime } = 0 \]

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

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

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

\[ {}y^{\prime } = y+{\mathrm e}^{-y}+{\mathrm e}^{-t} \]

\[ {}y^{\prime } = y^{3}+{\mathrm e}^{-5 t} \]

\[ {}y^{\prime } = \left (4 y+{\mathrm e}^{-t^{2}}\right ) {\mathrm e}^{2 y} \]

\[ {}y^{\prime } = {\mathrm e}^{-t}+\ln \left (1+y^{2}\right ) \]

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

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

\[ {}t^{3} y^{\prime \prime }-t y^{\prime }-\left (t^{2}+\frac {5}{4}\right ) y = 0 \]

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

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

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

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

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

\[ {}x^{3} \left (x^{2}+3\right ) y^{\prime \prime }+5 x y^{\prime }-\left (1+x \right ) y = 0 \]

\[ {}y^{\prime \prime }+\frac {y}{z^{3}} = 0 \]

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

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

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

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

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

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

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

\[ {}y^{\prime } = \operatorname {f0} \left (x \right )+\operatorname {f1} \left (x \right ) y+\operatorname {f2} \left (x \right ) y^{2}+\operatorname {f3} \left (x \right ) y^{3} \]

\[ {}y^{\prime } = f \left (x \right )+g \left (x \right ) y+h \left (x \right ) y^{n} \]

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

\[ {}y^{\prime }+f \left (x \right )+g \left (x \right ) \sin \left (a y\right )+h \left (x \right ) \cos \left (a y\right ) = 0 \]

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

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

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

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

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

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

\[ {}y y^{\prime }+f \left (x \right ) = g \left (x \right ) y \]

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

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

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

\[ {}\left (\operatorname {g0} \left (x \right )+y \operatorname {g1} \left (x \right )\right ) y^{\prime } = \operatorname {f0} \left (x \right )+\operatorname {f1} \left (x \right ) y+\operatorname {f2} \left (x \right ) y^{2}+\operatorname {f3} \left (x \right ) y^{3} \]

\[ {}\left (\cot \left (x \right )-2 y^{2}\right ) y^{\prime } = y^{3} \csc \left (x \right ) \sec \left (x \right ) \]

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

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

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

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

\[ {}x {y^{\prime }}^{3}-3 x^{2} y {y^{\prime }}^{2}+x \left (x^{5}+3 y^{2}\right ) y^{\prime }-2 x^{5} y-y^{3} = 0 \]

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

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

\[ {}x^{4} y^{\prime \prime }+x y^{\prime }+y = 0 \]

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

\[ {}y^{\prime \prime }+\frac {a y}{x^{\frac {3}{2}}} = 0 \]

\[ {}x^{3} y^{\prime \prime }+y = x^{\frac {3}{2}} \]

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

\[ {}s^{\prime } = t \ln \left (s^{2 t}\right )+8 t^{2} \]

\[ {}s^{2}+s^{\prime } = \frac {s+1}{s t} \]

\[ {}x^{\prime }+x t = {\mathrm e}^{x} \]

\[ {}x x^{\prime }+x t^{2} = \sin \left (t \right ) \]

\[ {}y y^{\prime }+2 x = 5 y^{3} \]

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

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

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

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

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

\[ {}x^{3} y^{\prime \prime }+y = \frac {1}{x^{4}} \]

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

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

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

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

\[ {}x^{3} y^{\prime \prime }+4 x^{2} y^{\prime }+3 y = 0 \]

\[ {}x^{3} \left (x^{2}-25\right ) \left (-2+x \right )^{2} y^{\prime \prime }+3 x \left (-2+x \right ) y^{\prime }+7 \left (x +5\right ) y = 0 \]

\[ {}x^{4} y^{\prime \prime }+\lambda y = 0 \]

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

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

\[ {}y^{\prime \prime \prime }-2 x y^{\prime \prime }+4 x^{2} y^{\prime }+8 x^{3} y = 0 \]

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

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

\[ {}x^{2} y^{\prime \prime }-5 y^{\prime }+3 x^{2} y = 0 \]

\[ {}\left [\begin {array}{c} y_{1}^{\prime }\left (x \right )=3 y_{1} \left (x \right )+x y_{3} \left (x \right ) \\ y_{2}^{\prime }\left (x \right )=y_{2} \left (x \right )+x^{3} y_{3} \left (x \right ) \\ y_{3}^{\prime }\left (x \right )=2 x y_{2} \left (x \right )-y_{2} \left (x \right )+{\mathrm e}^{x} y_{3} \left (x \right ) \end {array}\right ] \]

\[ {}x y^{\prime }+y = y^{\prime } \sqrt {1-x^{2} y} \]

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

\[ {}\sin \left (x \right ) \tan \left (y\right )+1+\cos \left (x \right ) \sec \left (x \right )^{2} y y^{\prime } = 0 \]

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

\[ {}x y y^{\prime \prime } = y^{\prime }+{y^{\prime }}^{3} \]

\[ {}x^{2} y^{\prime } = y \]

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

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

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

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

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

\[ {}\left [\begin {array}{c} x^{\prime }\left (t \right )=x \left (t \right ) y \left (t \right )+1 \\ y^{\prime }\left (t \right )=-x \left (t \right )+y \left (t \right ) \end {array}\right ] \]

\[ {}\left [\begin {array}{c} x^{\prime }\left (t \right )=t y \left (t \right )+1 \\ y^{\prime }\left (t \right )=-x \left (t \right ) t +y \left (t \right ) \end {array}\right ] \]

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

\[ {}y^{\prime \prime }+5 x y^{\prime }+\sqrt {x}\, y = 0 \]

\[ {}x^{3} y^{\prime \prime }+4 x^{2} y^{\prime }+3 y = 0 \]

\[ {}x^{3} \left (x^{2}-25\right ) \left (-2+x \right )^{2} y^{\prime \prime }+3 x \left (-2+x \right ) y^{\prime }+7 \left (x +5\right ) y = 0 \]

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

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

\[ {}y^{\prime } = \sqrt {1-x^{2}-y^{2}} \]

\[ {}y y^{\prime \prime } = x \]

\[ {}3 y y^{\prime \prime } = \sin \left (x \right ) \]

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

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

\[ {}y^{\prime \prime }-x^{3} y^{\prime }-x^{3} y-x^{4}-x^{3} = 0 \]

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

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

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

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

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

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

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

\[ {}2 x^{2} y^{\prime \prime }+2 x y^{\prime }-x y = 1 \]

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

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

\[ {}y^{\prime }+y = \frac {1}{x^{2}} \]

\[ {}y^{\prime } = \frac {1}{x} \]

\[ {}y^{\prime \prime } = \frac {1}{x} \]

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

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

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

\[ {}y^{\prime } = \frac {x y+3 x -2 y+6}{x y-3 x -2 y+6} \]

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

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

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

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

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

\[ {}y^{\prime }-f \left (x \right ) y^{n}-g \left (x \right ) y-h \left (x \right ) = 0 \]

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

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

\[ {}y^{\prime }+f \left (x \right ) \cos \left (a y\right )+g \left (x \right ) \sin \left (a y\right )+h \left (x \right ) = 0 \]

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

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

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

\[ {}f \left (x \right ) y^{\prime }+g \left (x \right ) s \left (y\right )+h \left (x \right ) = 0 \]

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

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

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

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

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

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

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

\[ {}\left (x^{2} y-1\right ) y^{\prime }+8 x y^{2}-8 = 0 \]

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

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

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

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

\[ {}{y^{\prime }}^{2}+2 f \left (x \right ) y y^{\prime }+g \left (x \right ) y^{2}+h \left (x \right ) = 0 \]

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

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

\[ {}\operatorname {d0} \left (x \right ) {y^{\prime }}^{2}+2 \operatorname {b0} \left (x \right ) y y^{\prime }+\operatorname {c0} \left (x \right ) y^{2}+2 \operatorname {d0} \left (x \right ) y^{\prime }+2 \operatorname {e0} \left (x \right ) y+\operatorname {f0} \left (x \right ) = 0 \]

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

\[ {}x y {y^{\prime }}^{2}+\left (x^{22}-y^{2}+a \right ) y^{\prime }-x y = 0 \]

\[ {}\left (\operatorname {b2} y+\operatorname {a2} x +\operatorname {c2} \right )^{2} {y^{\prime }}^{2}+\left (\operatorname {a1} x +\operatorname {b1} y+\operatorname {c1} \right ) y^{\prime }+\operatorname {b0} y+\operatorname {a0} +\operatorname {c0} = 0 \]

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

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

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

\[ {}{y^{\prime }}^{2}-\left (y^{4}+x y^{2}+x^{2}\right ) {y^{\prime }}^{2}+\left (x y^{6}+y^{4} x^{2}+x^{3} y^{2}\right ) y^{\prime }-x^{3} y^{6} = 0 \]

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

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

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

\[ {}\phi \left (f \left (x , y, y^{\prime }\right ), g \left (x , y, y^{\prime }\right )\right ) = 0 \]

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

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

\[ {}y^{\prime } = -\frac {1}{-\left (y^{3}\right )^{\frac {2}{3}} x -\textit {\_F1} \left (y^{3}-3 \ln \left (x \right )\right ) \left (y^{3}\right )^{\frac {1}{3}} x} \]

\[ {}y^{\prime } = -\frac {1}{-\ln \left (x \right ) \left (y^{3}\right )^{\frac {2}{3}}-\textit {\_F1} \left (y^{3}+3 \,\operatorname {expIntegral}_{1}\left (-\ln \left (x \right )\right )\right ) \ln \left (x \right ) \left (y^{3}\right )^{\frac {1}{3}}} \]

\[ {}y^{\prime } = -\frac {i \left (32 i x +64+64 y^{4}+32 x^{2} y^{2}+4 x^{4}+64 y^{6}+48 y^{4} x^{2}+12 x^{4} y^{2}+x^{6}\right )}{128 y} \]

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

\[ {}y^{\prime } = \frac {2 y^{6} \left (1+4 x y^{2}+y^{2}\right )}{y^{3}+4 y^{5} x +y^{5}+2+24 x y^{2}+96 y^{4} x^{2}+128 x^{3} y^{6}} \]

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

\[ {}y^{\prime \prime }-\left (n \left (n +1\right ) \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right )+B \right ) y = 0 \]

\[ {}y^{\prime \prime }-\left (\frac {p^{\prime \prime \prime \prime }\left (x \right )}{30}+\frac {7 p^{\prime \prime }\left (x \right )}{3}+a p \left (x \right )+b \right ) y = 0 \]

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

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

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

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

\[ {}y^{\prime \prime }+\frac {\left (11 \operatorname {WeierstrassP}\left (x , a , b\right ) \operatorname {WeierstrassPPrime}\left (x , a , b\right )-6 \operatorname {WeierstrassP}\left (x , a , b\right )^{2}+\frac {a}{2}\right ) y^{\prime }}{\operatorname {WeierstrassPPrime}\left (x , a , b\right )+\operatorname {WeierstrassP}\left (x , a , b\right )^{2}}+\frac {\left (\operatorname {WeierstrassPPrime}\left (x , a , b\right )^{2}-\operatorname {WeierstrassP}\left (x , a , b\right )^{2} \operatorname {WeierstrassPPrime}\left (x , a , b\right )-\operatorname {WeierstrassP}\left (x , a , b\right ) \left (6 \operatorname {WeierstrassP}\left (x , a , b\right )^{2}-\frac {a}{2}\right )\right ) y}{\operatorname {WeierstrassPPrime}\left (x , a , b\right )+\operatorname {WeierstrassP}\left (x , a , b\right )^{2}} = 0 \]

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

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

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

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

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

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

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

\[ {}x^{2} y^{\prime \prime }+\left (a \,x^{\operatorname {a1}}+b \right ) x y^{\prime }+\left (A \,x^{2 \operatorname {a1}}+B \,x^{\operatorname {a1}}+C \,x^{\operatorname {b1}}+\operatorname {DD} \right ) y = 0 \]

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

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

\[ {}y^{\prime \prime } = -\frac {\left (x^{2} \left (\left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right )+\left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )+\left (x^{2}-\operatorname {a3} \right ) \left (x^{2}-\operatorname {a1} \right )\right )-\left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )\right ) y^{\prime }}{x \left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )}-\frac {\left (A \,x^{2}+B \right ) y}{x \left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )} \]

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

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

\[ {}y^{\prime \prime } = \frac {2 \,\operatorname {JacobiSN}\left (x , k\right ) \operatorname {JacobiCN}\left (x , k\right ) \operatorname {JacobiDN}\left (x , k\right ) y^{\prime }-2 \left (1-2 \left (k^{2}+1\right ) \operatorname {JacobiSN}\left (a , k\right )^{2}+3 k^{2} \operatorname {JacobiSN}\left (a , k\right )^{4}\right ) y}{\operatorname {JacobiSN}\left (x , k\right )^{2}-\operatorname {JacobiSN}\left (a , k\right )} \]

\[ {}y^{\prime \prime } = -\frac {f^{\prime }\left (x \right ) y^{\prime }}{2 f \left (x \right )}-\frac {g \left (x \right ) y}{f \left (x \right )} \]

\[ {}y^{\prime \prime \prime }-3 \left (2 \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right )+a \right ) y^{\prime }+b y = 0 \]

\[ {}y^{\prime \prime \prime }+\left (-n^{2}+1\right ) \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y^{\prime }+\frac {\left (\left (-n^{2}+1\right ) \operatorname {WeierstrassPPrime}\left (x , \operatorname {g2} , \operatorname {g3}\right )-a \right ) y}{2} = 0 \]

\[ {}y^{\prime \prime \prime }-\left (4 n \left (n +1\right ) \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right )+a \right ) y^{\prime }-2 n \left (n +1\right ) \operatorname {WeierstrassPPrime}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y = 0 \]

\[ {}y^{\prime \prime \prime }+\left (A \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right )+a \right ) y^{\prime }+B \operatorname {WeierstrassPPrime}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y = 0 \]

\[ {}y^{\prime \prime \prime }-\left (3 k^{2} \operatorname {JacobiSN}\left (z , x\right )^{2}+a \right ) y^{\prime }+\left (b +c \operatorname {JacobiSN}\left (z , x\right )^{2}-3 k^{2} \operatorname {JacobiSN}\left (z , x\right ) \operatorname {JacobiCN}\left (z , x\right ) \operatorname {JacobiDN}\left (z , x\right )\right ) y = 0 \]

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

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

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

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

\[ {}27 y^{\prime \prime \prime }-36 n^{2} \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y^{\prime }-2 n \left (n +3\right ) \left (4 n -3\right ) \operatorname {WeierstrassPPrime}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y = 0 \]

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

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

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

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

\[ {}f^{\prime }\left (x \right ) y^{\prime \prime }+f \left (x \right ) y^{\prime \prime \prime }+g^{\prime }\left (x \right ) y^{\prime }+g \left (x \right ) y^{\prime \prime }+h^{\prime }\left (x \right ) y+h \left (x \right ) y^{\prime }+A \left (x \right ) \left (f \left (x \right ) y^{\prime \prime }+g \left (x \right ) y^{\prime }+h \left (x \right ) y\right ) = 0 \]

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

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

\[ {}y^{\prime \prime \prime \prime }+a \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y^{\prime \prime }+b \operatorname {WeierstrassPPrime}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y^{\prime }+\left (c \left (6 \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right )^{2}-\frac {\operatorname {g2}}{2}\right )+d \right ) y = 0 \]

\[ {}y^{\prime \prime \prime \prime }-\left (12 k^{2} \operatorname {JacobiSN}\left (z , x\right )^{2}+a \right ) y^{\prime \prime }+b y^{\prime }+\left (\alpha \operatorname {JacobiSN}\left (z , x\right )^{2}+\beta \right ) y = 0 \]

\[ {}y^{\left (5\right )}-a x y-b = 0 \]

\[ {}y^{\left (5\right )}+a \,x^{\nu } y^{\prime }+a \nu \,x^{\nu -1} y = 0 \]

\[ {}x y^{\left (5\right )}-\left (a A_{1} -A_{0} \right ) x -A_{1} -\left (\left (a A_{2} -A_{1} \right ) x +A_{2} \right ) y^{\prime } = 0 \]

\[ {}\left (x -a \right )^{5} \left (-b +x \right )^{5} y^{\left (5\right )}-c y = 0 \]

\[ {}y^{\prime \prime }-6 y^{2}-x = 0 \]

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

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

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

\[ {}y^{\prime \prime }+d +b x y+c y+a y^{3} = 0 \]

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

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

\[ {}y^{\prime \prime }+{\mathrm e}^{x} \sin \left (y\right ) = 0 \]

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

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

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

\[ {}y^{\prime \prime }-7 y^{\prime }-y^{\frac {3}{2}}+12 y = 0 \]

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

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

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

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

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

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

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

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

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

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

\[ {}y^{\prime \prime }+f \left (x , y\right ) y^{\prime }+g \left (x , y\right ) = 0 \]

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

\[ {}y^{\prime \prime }+a {y^{\prime }}^{2}+b y^{\prime }+c y = 0 \]

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

\[ {}y^{\prime \prime }-\frac {D\left (f \right )\left (y\right ) {y^{\prime }}^{3}}{f \left (y\right )}+g \left (x \right ) y^{\prime }+h \left (x \right ) f \left (y\right ) = 0 \]

\[ {}y^{\prime \prime }+\phi \left (y\right ) {y^{\prime }}^{2}+f \left (x \right ) y^{\prime }+g \left (x \right ) \Phi \left (y\right ) = 0 \]

\[ {}y^{\prime \prime }+f \left (y\right ) {y^{\prime }}^{2}+g \left (y\right ) y^{\prime }+h \left (y\right ) = 0 \]

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

\[ {}y^{\prime \prime }-k \,x^{a} y^{b} {y^{\prime }}^{r} = 0 \]

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

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

\[ {}y^{\prime \prime }-y f \left (x , \frac {y^{\prime }}{y}\right ) = 0 \]

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

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

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

\[ {}x y^{\prime \prime }+2 y^{\prime }+a \,x^{v} y^{n} = 0 \]

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

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

\[ {}x y^{\prime \prime }+a y^{\prime }+b \,x^{5-2 a} {\mathrm e}^{y} = 0 \]

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

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

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

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

\[ {}4 x^{2} y^{\prime \prime }-x^{4} {y^{\prime }}^{2}+4 y = 0 \]

\[ {}x^{3} \left (y^{\prime \prime }+y y^{\prime }-y^{3}\right )+12 x y+24 = 0 \]

\[ {}2 x^{3} y^{\prime \prime }+x^{2} \left (9+2 x y\right ) y^{\prime }+b +x y \left (a +3 x y-2 x^{2} y^{2}\right ) = 0 \]

\[ {}2 \left (-x^{k}+4 x^{3}\right ) \left (y^{\prime \prime }+y y^{\prime }-y^{3}\right )-\left (k \,x^{k -1}-12 x^{2}\right ) \left (3 y^{\prime }+y^{2}\right )+a x y+b = 0 \]

\[ {}x^{4} y^{\prime \prime }+a^{2} y^{n} = 0 \]

\[ {}y^{\prime \prime } \sqrt {x}-y^{\frac {3}{2}} = 0 \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}2 y y^{\prime \prime }-{y^{\prime }}^{2}+1+2 x y^{2}+a y^{3} = 0 \]

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

\[ {}2 y y^{\prime \prime }-{y^{\prime }}^{2}+b -4 \left (x^{2}+a \right ) y^{2}-8 x y^{3}-3 y^{4} = 0 \]

\[ {}2 y y^{\prime \prime }-{y^{\prime }}^{2}+3 f \left (x \right ) y y^{\prime }+2 \left (f \left (x \right )^{2}+f^{\prime }\left (x \right )\right ) y^{2}-8 y^{3} = 0 \]

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

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

\[ {}4 y y^{\prime \prime }-3 {y^{\prime }}^{2}+\left (6 y^{2}-\frac {2 f^{\prime }\left (x \right ) y}{f \left (x \right )}\right ) y^{\prime }+y^{4}-2 y^{2} y^{\prime }+g \left (x \right ) y^{2}+f \left (x \right ) y = 0 \]

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

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

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

\[ {}\operatorname {f0} \left (x \right ) y y^{\prime \prime }+\operatorname {f1} \left (x \right ) {y^{\prime }}^{2}+\operatorname {f2} \left (x \right ) y y^{\prime }+\operatorname {f3} \left (x \right ) y^{2} = 0 \]

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

\[ {}-2 y \left (1-y\right ) y^{\prime \prime }+\left (-3 y+1\right ) {y^{\prime }}^{2}-4 y y^{\prime } \left (f \left (x \right ) y+g \left (x \right )\right )+\left (1-y\right )^{3} \left (\operatorname {f0} \left (x \right )^{2} y^{2}-\operatorname {f1} \left (x \right )^{2}\right )+4 y^{2} \left (1-y\right ) \left (f \left (x \right )^{2}-g \left (x \right )^{2}-g^{\prime }\left (x \right )-f^{\prime }\left (x \right )\right ) = 0 \]

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

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

\[ {}2 y^{3} y^{\prime \prime }+y^{2} {y^{\prime }}^{2}-x^{2} a -b x -c = 0 \]

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

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

\[ {}h \left (y\right ) y^{\prime \prime }-D\left (h \right )\left (y\right ) {y^{\prime }}^{2}-h \left (y\right )^{2} j \left (x , \frac {y^{\prime }}{h \left (y\right )}\right ) = 0 \]

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

\[ {}\left (\operatorname {f1} y^{\prime }+\operatorname {f2} y\right ) y^{\prime \prime }+\operatorname {f3} {y^{\prime }}^{2}+\operatorname {f4} \left (x \right ) y y^{\prime }+\operatorname {f5} \left (x \right ) y^{2} = 0 \]

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

\[ {}h \left (y^{\prime }\right ) y^{\prime \prime }+j \left (y\right ) y^{\prime }+f = 0 \]

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

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

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

\[ {}\left (2 y y^{\prime \prime }-{y^{\prime }}^{2}\right )^{3}+32 y^{\prime \prime } \left (x y^{\prime \prime }-y^{\prime }\right )^{3} = 0 \]

\[ {}\sqrt {a {y^{\prime \prime }}^{2}+b {y^{\prime }}^{2}}+c y y^{\prime \prime }+d {y^{\prime }}^{2} = 0 \]

\[ {}y^{\prime \prime \prime }+y y^{\prime \prime }-{y^{\prime }}^{2}+1 = 0 \]

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

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

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

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

\[ {}y^{\prime } \left (f^{\prime \prime \prime }\left (x \right ) y^{\prime }+3 f^{\prime \prime }\left (x \right ) y^{\prime \prime }+3 f^{\prime }\left (x \right ) y^{\prime \prime \prime }+f \left (x \right ) y^{\prime \prime \prime \prime }\right )-y^{\prime \prime } f y^{\prime \prime \prime }+{y^{\prime }}^{3} \left (f^{\prime }\left (x \right ) y^{\prime }+f \left (x \right ) y^{\prime \prime }\right )+2 q \left (x \right ) {y^{\prime }}^{2} \sin \left (y\right )+\left (q \left (x \right ) y^{\prime \prime }-q^{\prime }\left (x \right ) y^{\prime }\right ) \cos \left (y\right ) = 0 \]

\[ {}y^{\prime \prime \prime } = f \left (y\right ) \]

\[ {}\left [\begin {array}{c} x^{\prime \prime }\left (t \right )=\left (3 \cos \left (a t +b \right )^{2}-1\right ) c^{2} x \left (t \right )+\frac {3 c^{2} y \left (t \right ) \sin \left (2 a t b \right )}{2} \\ y^{\prime \prime }\left (t \right )=\left (3 \sin \left (a t +b \right )^{2}-1\right ) c^{2} y \left (t \right )+\frac {3 c^{2} x \left (t \right ) \sin \left (2 a t b \right )}{2} \end {array}\right ] \]

\[ {}\left [\begin {array}{c} x^{\prime }\left (t \right )=h \left (t \right ) y \left (t \right )-g \left (t \right ) z \left (t \right ) \\ y^{\prime }\left (t \right )=f \left (t \right ) z \left (t \right )-h \left (t \right ) x \left (t \right ) \\ z^{\prime }\left (t \right )=x \left (t \right ) g \left (t \right )-y \left (t \right ) f \left (t \right ) \end {array}\right ] \]

\[ {}\left [\begin {array}{c} x^{\prime }\left (t \right )=x \left (t \right ) \left (a \left (p x \left (t \right )+q y \left (t \right )\right )+\alpha \right ) \\ y^{\prime }\left (t \right )=y \left (t \right ) \left (\beta +b \left (p x \left (t \right )+q y \left (t \right )\right )\right ) \end {array}\right ] \]

\[ {}\left [\begin {array}{c} x^{\prime }\left (t \right )=-x \left (t \right ) y \left (t \right )^{2}+x \left (t \right )+y \left (t \right ) \\ y^{\prime }\left (t \right )=x \left (t \right )^{2} y \left (t \right )-x \left (t \right )-y \left (t \right ) \end {array}\right ] \]

\[ {}\left [\begin {array}{c} x^{\prime }\left (t \right )=x \left (t \right )+y \left (t \right )-x \left (t \right ) \left (x \left (t \right )^{2}+y \left (t \right )^{2}\right ) \\ y^{\prime }\left (t \right )=-x \left (t \right )+y \left (t \right )-y \left (t \right ) \left (x \left (t \right )^{2}+y \left (t \right )^{2}\right ) \end {array}\right ] \]

\[ {}\left [\begin {array}{c} x^{\prime }\left (t \right )=-y \left (t \right )+x \left (t \right ) \left (x \left (t \right )^{2}+y \left (t \right )^{2}-1\right ) \\ y^{\prime }\left (t \right )=x \left (t \right )+y \left (t \right ) \left (x \left (t \right )^{2}+y \left (t \right )^{2}-1\right ) \end {array}\right ] \]

\[ {}\left [\begin {array}{c} x^{\prime }\left (t \right )=-y \left (t \right ) \left (x \left (t \right )^{2}+y \left (t \right )^{2}\right ) \\ y^{\prime }\left (t \right )=\left \{\begin {array}{cc} x \left (t \right )^{2}+y \left (t \right )^{2} & 2 x \left (t \right )\le x \left (t \right )^{2}+y \left (t \right )^{2} \\ \left (\frac {x \left (t \right )}{2}-\frac {y \left (t \right )^{2}}{2 x \left (t \right )}\right ) \left (x \left (t \right )^{2}+y \left (t \right )^{2}\right ) & \operatorname {otherwise} \end {array}\right . \end {array}\right ] \]

\[ {}\left [\begin {array}{c} x^{\prime }\left (t \right )=-y \left (t \right )+\left \{\begin {array}{cc} x \left (t \right ) \left (x \left (t \right )^{2}+y \left (t \right )^{2}-1\right ) \sin \left (\frac {1}{x \left (t \right )^{2}+y \left (t \right )^{2}}\right ) & x \left (t \right )^{2}+y \left (t \right )^{2}\neq 1 \\ 0 & \operatorname {otherwise} \end {array}\right . \\ y^{\prime }\left (t \right )=x \left (t \right )+\left \{\begin {array}{cc} y \left (t \right ) \left (x \left (t \right )^{2}+y \left (t \right )^{2}-1\right ) \sin \left (\frac {1}{x \left (t \right )^{2}+y \left (t \right )^{2}}\right ) & x \left (t \right )^{2}+y \left (t \right )^{2}\neq 1 \\ 0 & \operatorname {otherwise} \end {array}\right . \end {array}\right ] \]

\[ {}\left [\begin {array}{c} x^{\prime \prime }\left (t \right )=a \,{\mathrm e}^{2 x \left (t \right )}-{\mathrm e}^{-x \left (t \right )}+{\mathrm e}^{-2 x \left (t \right )} \cos \left (y \left (t \right )\right )^{2} \\ y^{\prime \prime }\left (t \right )={\mathrm e}^{-2 x \left (t \right )} \sin \left (y \left (t \right )\right ) \cos \left (y \left (t \right )\right )-\frac {\sin \left (y \left (t \right )\right )}{\cos \left (y \left (t \right )\right )^{3}} \end {array}\right ] \]

\[ {}\left [\begin {array}{c} x^{\prime \prime }\left (t \right )=\frac {k x \left (t \right )}{\left (x \left (t \right )^{2}+y \left (t \right )^{2}\right )^{\frac {3}{2}}} \\ y^{\prime \prime }\left (t \right )=\frac {k y \left (t \right )}{\left (x \left (t \right )^{2}+y \left (t \right )^{2}\right )^{\frac {3}{2}}} \end {array}\right ] \]

\[ {}\left [\begin {array}{c} x^{\prime }\left (t \right )=x \left (t \right ) \left (y \left (t \right )-z \left (t \right )\right ) \\ y^{\prime }\left (t \right )=y \left (t \right ) \left (-x \left (t \right )+z \left (t \right )\right ) \\ z^{\prime }\left (t \right )=z \left (t \right ) \left (x \left (t \right )-y \left (t \right )\right ) \end {array}\right ] \]

\[ {}\left [\begin {array}{c} x^{\prime }\left (t \right )=\frac {x \left (t \right )^{2}}{2}-\frac {y \left (t \right )}{24} \\ y^{\prime }\left (t \right )=2 x \left (t \right ) y \left (t \right )-3 z \left (t \right ) \\ z^{\prime }\left (t \right )=3 x \left (t \right ) z \left (t \right )-\frac {y \left (t \right )^{2}}{6} \end {array}\right ] \]

\[ {}\left [\begin {array}{c} x^{\prime }\left (t \right )=x \left (t \right ) \left (y \left (t \right )^{2}-z \left (t \right )^{2}\right ) \\ y^{\prime }\left (t \right )=y \left (t \right ) \left (z \left (t \right )^{2}-x \left (t \right )^{2}\right ) \\ z^{\prime }\left (t \right )=z \left (t \right ) \left (x \left (t \right )^{2}-y \left (t \right )^{2}\right ) \end {array}\right ] \]

\[ {}\left [\begin {array}{c} x^{\prime }\left (t \right )=-x \left (t \right ) y \left (t \right )^{2}+x \left (t \right )+y \left (t \right ) \\ y^{\prime }\left (t \right )=x \left (t \right )^{2} y \left (t \right )-x \left (t \right )-y \left (t \right ) \\ z^{\prime }\left (t \right )=y \left (t \right )^{2}-x \left (t \right )^{2} \end {array}\right ] \]

\[ {}\left [\begin {array}{c} x_{1}^{\prime }\left (t \right ) \sin \left (x_{2} \left (t \right )\right )=x_{4} \left (t \right ) \sin \left (x_{3} \left (t \right )\right )+x_{5} \left (t \right ) \cos \left (x_{3} \left (t \right )\right ) \\ x_{2}^{\prime }\left (t \right )=x_{4} \left (t \right ) \cos \left (x_{3} \left (t \right )\right )-x_{5} \left (t \right ) \sin \left (x_{3} \left (t \right )\right ) \\ x_{3}^{\prime }\left (t \right )+x_{1}^{\prime }\left (t \right ) \cos \left (x_{2} \left (t \right )\right )=a \\ x_{4}^{\prime }\left (t \right )-\left (1-\lambda \right ) a x_{5} \left (t \right )=-m \sin \left (x_{2} \left (t \right )\right ) \cos \left (x_{3} \left (t \right )\right ) \\ x_{5}^{\prime }\left (t \right )+\left (1-\lambda \right ) a x_{4} \left (t \right )=m \sin \left (x_{2} \left (t \right )\right ) \sin \left (x_{3} \left (t \right )\right ) \end {array}\right ] \]

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

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

\[ {}y^{\prime } = y^{2}-2 \lambda ^{2} \tan \left (x \right )^{2}-2 \lambda ^{2} \cot \left (\lambda x \right )^{2} \]

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

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

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

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

\[ {}y y^{\prime }-y = A \,x^{k -1}-k B \,x^{k}+k \,B^{2} x^{2 k -1} \]

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

\[ {}y y^{\prime }-y = \frac {2 m -2}{\left (m -3\right )^{2}}+\frac {2 A \left (m \left (3+m \right ) \sqrt {x}+\left (4 m^{2}+3 m +9\right ) A +\frac {3 m \left (3+m \right ) A^{2}}{\sqrt {x}}\right )}{\left (m -3\right )^{2}} \]

\[ {}y y^{\prime }-y = -\frac {3 x}{16}+\frac {5 A}{x^{\frac {1}{3}}}-\frac {12 A^{2}}{x^{\frac {5}{3}}} \]

\[ {}y y^{\prime }-y = -\frac {6 X}{25}+\frac {2 A \left (2 \sqrt {x}+19 A +\frac {6 A^{2}}{\sqrt {x}}\right )}{25} \]

\[ {}y y^{\prime }-y = \frac {3 x}{8}+\frac {3 \sqrt {a^{2}+x^{2}}}{8}-\frac {a^{2}}{16 \sqrt {a^{2}+x^{2}}} \]

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

\[ {}y y^{\prime }-y = -\frac {12 x}{49}+\frac {6 A \left (-3 \sqrt {x}+23 A +\frac {12 A^{2}}{\sqrt {x}}\right )}{49} \]

\[ {}y y^{\prime }-y = -\frac {30 x}{121}+\frac {3 A \left (21 \sqrt {x}+35 A +\frac {6 A^{2}}{\sqrt {x}}\right )}{242} \]

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

\[ {}y y^{\prime }-y = -\frac {12 x}{49}+\frac {4 A \left (-10 \sqrt {x}+27 A +\frac {10 A^{2}}{\sqrt {x}}\right )}{49} \]

\[ {}y y^{\prime }-y = -\frac {x}{4}+\frac {6 A \left (\sqrt {x}+8 A +\frac {5 A^{2}}{\sqrt {x}}\right )}{49} \]

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

\[ {}y y^{\prime }-y = \frac {3 x}{8}+\frac {3 \sqrt {b^{2}+x^{2}}}{8}+\frac {3 b^{2}}{16 \sqrt {b^{2}+x^{2}}} \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}\left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\left (\alpha \gamma +\beta \right ) x +\beta \lambda \right ) y^{\prime }-\left (\alpha x +\beta \right ) y = 0 \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}{x^{\prime }}^{2}+x t = \sqrt {t +1} \]

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

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

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

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

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

\[ {}\left (x^{5}+x^{4}-6 x^{3}\right ) y^{\prime \prime }+x^{2} y^{\prime }+\left (-2+x \right ) y = 0 \]

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

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

\[ {}y^{\prime } = \sin \left (x y\right ) \]

\[ {}y^{\prime } = t \ln \left (y^{2 t}\right )+t^{2} \]

\[ {}y^{\prime } = \ln \left (x y\right ) \]

\[ {}y^{\prime \prime }+y y^{\prime \prime \prime \prime } = 1 \]

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

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

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

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

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

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

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

\[ {}\sin \left (x \right ) y^{\prime \prime }+x y^{\prime }+7 y = 1 \]

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

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

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

\[ {}t^{2} y^{\prime \prime }-6 t y^{\prime }+\sin \left (2 t \right ) y = \ln \left (t \right ) \]

\[ {}y^{\prime \prime }+t y^{\prime }-y \ln \left (t \right ) = \cos \left (2 t \right ) \]

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

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

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

\[ {}x^{\prime \prime }+x^{\prime }+x+x^{3} = 0 \]

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

\[ {}y^{\prime } = x^{3}+y^{3} \]

\[ {}y^{\prime } = \frac {1}{\sqrt {15-x^{2}-y^{2}}} \]

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

\[ {}\left [\begin {array}{c} y_{1}^{\prime }\left (x \right )=\sin \left (x \right ) y_{1} \left (x \right )+\sqrt {x}\, y_{2} \left (x \right )+\ln \left (x \right ) \\ y_{2}^{\prime }\left (x \right )=\tan \left (x \right ) y_{1} \left (x \right )-{\mathrm e}^{x} y_{2} \left (x \right )+1 \end {array}\right ] \]

\[ {}\left [\begin {array}{c} y_{1}^{\prime }\left (x \right )=\sin \left (x \right ) y_{1} \left (x \right )+\sqrt {x}\, y_{2} \left (x \right )+\ln \left (x \right ) \\ y_{2}^{\prime }\left (x \right )=\tan \left (x \right ) y_{1} \left (x \right )-{\mathrm e}^{x} y_{2} \left (x \right )+1 \end {array}\right ] \]

\[ {}\left [\begin {array}{c} y_{1}^{\prime }\left (x \right )={\mathrm e}^{-x} y_{1} \left (x \right )-\sqrt {1+x}\, y_{2} \left (x \right )+x^{2} \\ y_{2}^{\prime }\left (x \right )=\frac {y_{1} \left (x \right )}{\left (-2+x \right )^{2}} \end {array}\right ] \]

\[ {}\left [\begin {array}{c} y_{1}^{\prime }\left (x \right )={\mathrm e}^{-x} y_{1} \left (x \right )-\sqrt {1+x}\, y_{2} \left (x \right )+x^{2} \\ y_{2}^{\prime }\left (x \right )=\frac {y_{1} \left (x \right )}{\left (-2+x \right )^{2}} \end {array}\right ] \]

\[ {}\left [\begin {array}{c} y_{1}^{\prime }\left (x \right )=2 x y_{1} \left (x \right )-x^{2} y_{2} \left (x \right )+4 x \\ y_{2}^{\prime }\left (x \right )={\mathrm e}^{x} y_{1} \left (x \right )+3 \,{\mathrm e}^{-x} y_{2} \left (x \right )-\cos \left (3 x \right ) \end {array}\right ] \]

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

\[ {}y^{\prime } = \left (y-3\right ) \left (\sin \left (y\right ) \sin \left (t \right )+\cos \left (t \right )+1\right ) \]

\[ {}y^{\prime } = \left (y-1\right ) \left (y-2\right ) \left (y-{\mathrm e}^{\frac {t}{2}}\right ) \]

\[ {}y^{2} y^{\prime \prime } = 8 x^{2} \]

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

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

\[ {}y^{\prime \prime }+x^{2} y^{\prime }+4 y = y^{3} \]

\[ {}y y^{\prime \prime \prime }+6 y^{\prime \prime }+3 y^{\prime } = y \]

\[ {}x^{3} y^{\prime \prime \prime }-4 y^{\prime \prime }+10 y^{\prime }-12 y = 0 \]

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

\[ {}x y^{\prime \prime }-y^{\prime }+4 x^{3} y = 0 \]

\[ {}\left [\begin {array}{c} x^{\prime }\left (t \right )=x \left (t \right ) y \left (t \right )-6 y \left (t \right ) \\ y^{\prime }\left (t \right )=x \left (t \right )-y \left (t \right )-5 \end {array}\right ] \]

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

\[ {}x^{\prime \prime }+2 \sin \left (x\right ) = \sin \left (2 t \right ) \]

\[ {}4 x \left (x^{2}+y^{2}\right )-5 y+4 y \left (x^{2}+y^{2}-5 x \right ) y^{\prime } = 0 \]

\[ {}y^{\prime \prime }+4 y = t \]

\[ {}y^{\prime }+t^{2} = \frac {1}{y^{2}} \]

\[ {}y^{\prime } = \sin \left (y\right )-\cos \left (t \right ) \]

\[ {}{\mathrm e}^{2 t}-y-\left ({\mathrm e}^{y}-t \right ) y^{\prime } = 0 \]

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

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

\[ {}\frac {2 t}{t^{2}+1}+y+\left ({\mathrm e}^{y}+t \right ) y^{\prime } = 0 \]

\[ {}y^{\prime } = x +y^{\frac {1}{3}} \]

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

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

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

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

\[ {}x \left (1+x \right ) y^{\prime \prime }+\frac {y^{\prime }}{x^{2}}+5 y = 0 \]

\[ {}y^{\prime \prime }-8 y^{\prime }+16 y = \frac {{\mathrm e}^{4 t}}{t^{3}} \]

\[ {}y^{\prime } = \sin \left (y\right )-\cos \left (x \right ) \]

\[ {}y^{\prime \prime \prime } = \sqrt {1-{y^{\prime \prime }}^{2}} \]

\[ {}4 x y^{\prime \prime }+2 y^{\prime }+y = 1 \]

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

\[ {}x^{\prime \prime }+{\mathrm e}^{-x^{\prime }}-x = 0 \]

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

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

\[ {}\left [\begin {array}{c} x^{\prime }\left (t \right )=\cos \left (x \left (t \right )\right )^{2} \cos \left (y \left (t \right )\right )^{2}+\sin \left (x \left (t \right )\right )^{2} \cos \left (y \left (t \right )\right )^{2} \\ y^{\prime }\left (t \right )=-\frac {\sin \left (2 x \left (t \right )\right ) \sin \left (2 y \left (t \right )\right )}{2} \end {array}\right ] \]