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

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

\[ {}y^{\prime } = -\frac {216 y \left (-2 y^{4}-3 y^{3}-6 y^{2}-6 y+6 x +6\right )}{-1296 x y+4428 y^{5}-1944 x y^{2}+2808 y^{4}-1296 y^{2}+216 x^{3}-1296 y+594 y^{7}+1728 y^{3}-648 x^{2} y-18 y^{8}-315 y^{9}-432 y^{4} x -126 y^{10}-8 y^{12}-36 y^{11}+2484 y^{6}-648 x^{2} y^{2}-216 y^{4} x^{2}+594 x y^{6}+1080 y^{5} x +72 y^{8} x +216 y^{7} x -324 x^{2} y^{3}-648 x y^{3}} \]

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

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

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

\[ {}y^{\prime } = y^{3}-3 x^{2} y^{2}+3 y x^{4}-x^{6}+2 x \]

\[ {}y^{\prime } = y^{3}+x^{2} y^{2}+\frac {y x^{4}}{3}+\frac {x^{6}}{27}-\frac {2 x}{3} \]

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

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

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

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

\[ {}y^{\prime } = \frac {y^{3} x^{3}+6 x^{2} y^{2}+12 x y+8+2 x}{x^{3}} \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x +\operatorname {c1} \right ) y = 0 \]

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

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

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

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

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

\[ {}y^{\prime \prime }+y^{\prime } \sqrt {x}+\left (\frac {1}{4 \sqrt {x}}+\frac {x}{4}-9\right ) y-x \,{\mathrm e}^{-\frac {x^{\frac {3}{2}}}{3}} = 0 \]

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

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

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

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

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

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

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