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

\[ {}y^{\prime } = \frac {y^{2}}{y^{2}+y^{\frac {3}{2}}+\sqrt {y}\, x^{2}-2 y^{\frac {3}{2}} x +y^{\frac {5}{2}}+x^{3}-3 x^{2} y+3 x y^{2}-y^{3}} \]

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

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

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

\[ {}y^{\prime } = \frac {-8 x^{2} y^{3}+16 x y^{2}+16 x y^{3}-8+12 x y-6 x^{2} y^{2}+x^{3} y^{3}}{16 \left (-2+x y-2 y\right ) x} \]

\[ {}y^{\prime } = -\frac {\left (-8 \,{\mathrm e}^{-x^{2}}+8 x^{2} {\mathrm e}^{-x^{2}}-8-8 y^{2}+8 x^{2} {\mathrm e}^{-x^{2}} y-2 x^{4} {\mathrm e}^{-2 x^{2}}-8 y^{3}+12 x^{2} {\mathrm e}^{-x^{2}} y^{2}-6 y x^{4} {\mathrm e}^{-2 x^{2}}+x^{6} {\mathrm e}^{-3 x^{2}}\right ) x}{8} \]

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

\[ {}y^{\prime } = -\frac {16 x y^{3}-8 y^{3}-8 y+8 x y^{2}-2 x^{2} y^{3}-8+12 x y-6 x^{2} y^{2}+x^{3} y^{3}}{32 y x} \]

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

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

\[ {}y^{\prime } = \frac {\left (27 y^{3}+27 \,{\mathrm e}^{3 x^{2}} y+18 \,{\mathrm e}^{3 x^{2}} y^{2}+3 y^{3} {\mathrm e}^{3 x^{2}}+27 \,{\mathrm e}^{\frac {9 x^{2}}{2}}+27 \,{\mathrm e}^{\frac {9 x^{2}}{2}} y+9 \,{\mathrm e}^{\frac {9 x^{2}}{2}} y^{2}+{\mathrm e}^{\frac {9 x^{2}}{2}} y^{3}\right ) {\mathrm e}^{3 x^{2}} x \,{\mathrm e}^{-\frac {9 x^{2}}{2}}}{243 y} \]

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

\[ {}y^{\prime } = \frac {x}{2}+1+y^{2}+\frac {x^{2} y}{4}-x y-\frac {x^{4}}{8}+\frac {x^{3}}{8}+\frac {x^{2}}{4}+y^{3}-\frac {3 x^{2} y^{2}}{4}-\frac {3 x y^{2}}{2}+\frac {3 y x^{4}}{16}+\frac {3 x^{3} y}{4}-\frac {x^{6}}{64}-\frac {3 x^{5}}{32} \]

\[ {}y^{\prime } = -\frac {x}{2}+1+y^{2}+\frac {7 x^{2} y}{2}-2 x y+\frac {13 x^{4}}{16}-\frac {3 x^{3}}{2}+x^{2}+y^{3}+\frac {3 x^{2} y^{2}}{4}-3 x y^{2}+\frac {3 y x^{4}}{16}-\frac {3 x^{3} y}{2}+\frac {x^{6}}{64}-\frac {3 x^{5}}{16} \]

\[ {}y^{\prime } = -\frac {x}{4}+1+y^{2}+\frac {7 x^{2} y}{16}-\frac {x y}{2}+\frac {5 x^{4}}{128}-\frac {5 x^{3}}{64}+\frac {x^{2}}{16}+y^{3}+\frac {3 x^{2} y^{2}}{8}-\frac {3 x y^{2}}{4}+\frac {3 y x^{4}}{64}-\frac {3 x^{3} y}{16}+\frac {x^{6}}{512}-\frac {3 x^{5}}{256} \]

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

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

\[ {}y^{\prime } = \frac {-32 x y+16 x^{3}+16 x^{2}-32 x -64 y^{3}+48 x^{2} y^{2}+96 x y^{2}-12 y x^{4}-48 x^{3} y-48 x^{2} y+x^{6}+6 x^{5}+12 x^{4}}{-64 y+16 x^{2}+32 x -64} \]

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

\[ {}y^{\prime } = \frac {-32 x y-72 x^{3}+32 x^{2}-32 x +64 y^{3}+48 x^{2} y^{2}-192 x y^{2}+12 y x^{4}-96 x^{3} y+192 x^{2} y+x^{6}-12 x^{5}+48 x^{4}}{64 y+16 x^{2}-64 x +64} \]

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

\[ {}y^{\prime } = \frac {-128 x y-24 x^{3}+32 x^{2}-128 x +512 y^{3}+192 x^{2} y^{2}-384 x y^{2}+24 y x^{4}-96 x^{3} y+96 x^{2} y+x^{6}-6 x^{5}+12 x^{4}}{512 y+64 x^{2}-128 x +512} \]

\[ {}y^{\prime } = \frac {-32 y a x -8 a^{2} x^{3}-16 a b \,x^{2}-32 a x +64 y^{3}+48 x^{2} a y^{2}+96 y^{2} b x +12 y a^{2} x^{4}+48 y a \,x^{3} b +48 y b^{2} x^{2}+a^{3} x^{6}+6 a^{2} x^{5} b +12 a \,x^{4} b^{2}+8 b^{3} x^{3}}{64 y+16 a \,x^{2}+32 b x +64} \]

\[ {}y^{\prime } = \frac {-32 x y-8 x^{3}-16 a \,x^{2}-32 x +64 y^{3}+48 x^{2} y^{2}+96 a x y^{2}+12 y x^{4}+48 y a \,x^{3}+48 a^{2} x^{2} y+x^{6}+6 x^{5} a +12 a^{2} x^{4}+8 a^{3} x^{3}}{64 y+16 x^{2}+32 a x +64} \]

\[ {}y^{\prime } = \frac {\left (-8 \,{\mathrm e}^{-x^{2}} y+4 x^{2} {\mathrm e}^{-2 x^{2}}-8 \,{\mathrm e}^{-x^{2}}+8 x^{2} {\mathrm e}^{-x^{2}} y-4 x^{4} {\mathrm e}^{-2 x^{2}}+8 x^{2} {\mathrm e}^{-x^{2}}-8 y^{3}+12 x^{2} {\mathrm e}^{-x^{2}} y^{2}-6 y x^{4} {\mathrm e}^{-2 x^{2}}+x^{6} {\mathrm e}^{-3 x^{2}}\right ) x}{-8 y+4 x^{2} {\mathrm e}^{-x^{2}}-8} \]

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

\[ {}y^{\prime } = -\frac {216 y}{-216 y^{4}-252 y^{3}-396 y^{2}-216 y+36 x^{2}-72 x y+60 y^{5}-36 x y^{3}-72 x y^{2}-24 x y^{4}+4 y^{8}+12 y^{7}+33 y^{6}} \]

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

\[ {}y^{\prime } = -\frac {a x}{2}+1+y^{2}+\frac {a \,x^{2} y}{2}+b x y+\frac {a^{2} x^{4}}{16}+\frac {a \,x^{3} b}{4}+\frac {b^{2} x^{2}}{4}+y^{3}+\frac {3 x^{2} a y^{2}}{4}+\frac {3 y^{2} b x}{2}+\frac {3 y a^{2} x^{4}}{16}+\frac {3 y a \,x^{3} b}{4}+\frac {3 y b^{2} x^{2}}{4}+\frac {a^{3} x^{6}}{64}+\frac {3 a^{2} x^{5} b}{32}+\frac {3 a \,x^{4} b^{2}}{16}+\frac {b^{3} x^{3}}{8} \]

\[ {}y^{\prime } = -\frac {x}{2}+1+y^{2}+\frac {x^{2} y}{2}+y a x +\frac {x^{4}}{16}+\frac {a \,x^{3}}{4}+\frac {a^{2} x^{2}}{4}+y^{3}+\frac {3 x^{2} y^{2}}{4}+\frac {3 a x y^{2}}{2}+\frac {3 y x^{4}}{16}+\frac {3 y a \,x^{3}}{4}+\frac {3 a^{2} x^{2} y}{4}+\frac {x^{6}}{64}+\frac {3 x^{5} a}{32}+\frac {3 a^{2} x^{4}}{16}+\frac {a^{3} x^{3}}{8} \]

\[ {}y^{\prime } = -\frac {-y+\sqrt {x^{2}+y^{2}}\, x^{2}-x \sqrt {x^{2}+y^{2}}\, y+x^{4} \sqrt {x^{2}+y^{2}}-x^{3} \sqrt {x^{2}+y^{2}}\, y+x^{5} \sqrt {x^{2}+y^{2}}-x^{4} \sqrt {x^{2}+y^{2}}\, y}{x} \]

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

\[ {}y^{\prime } = \frac {150 x^{3}+125 \sqrt {x}+125+125 y^{2}-100 x^{3} y-500 y \sqrt {x}+20 x^{6}+200 x^{\frac {7}{2}}+500 x +125 y^{3}-150 x^{3} y^{2}-750 y^{2} \sqrt {x}+60 y x^{6}+600 y x^{\frac {7}{2}}+1500 x y-8 x^{9}-120 x^{\frac {13}{2}}-600 x^{4}-1000 x^{\frac {3}{2}}}{125 x} \]

\[ {}y^{\prime } = \frac {-150 x^{3} y+60 x^{6}+350 x^{\frac {7}{2}}-150 x^{3}-125 y \sqrt {x}+250 x -125 \sqrt {x}-125 y^{3}+150 x^{3} y^{2}+750 y^{2} \sqrt {x}-60 y x^{6}-600 y x^{\frac {7}{2}}-1500 x y+8 x^{9}+120 x^{\frac {13}{2}}+600 x^{4}+1000 x^{\frac {3}{2}}}{25 \left (-5 y+2 x^{3}+10 \sqrt {x}-5\right ) x} \]

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

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

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

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

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

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

\[ {}y^{\prime } = \frac {4 x \left (a -1\right ) \left (a +1\right ) \left (-y^{2}+a^{2} x^{2}-x^{2}-2\right )}{-4 y^{3}+4 a^{2} x^{2} y-4 x^{2} y-8 y-a^{2} y^{6}+3 a^{4} y^{4} x^{2}-6 y^{4} a^{2} x^{2}-3 a^{6} y^{2} x^{4}+9 y^{2} a^{4} x^{4}-9 y^{2} a^{2} x^{4}+a^{8} x^{6}-4 a^{6} x^{6}+6 a^{4} x^{6}-4 a^{2} x^{6}+y^{6}+3 x^{2} y^{4}+3 x^{4} y^{2}+x^{6}} \]

\[ {}y^{\prime } = \frac {-4 \cos \relax (x ) x +4 \sin \relax (x ) x^{2}+4 x +4+4 y^{2}+8 y \cos \relax (x ) x -8 x y+2 x^{2} \cos \left (2 x \right )+6 x^{2}-8 x^{2} \cos \relax (x )+4 y^{3}+12 y^{2} \cos \relax (x ) x -12 x y^{2}+6 y x^{2} \cos \left (2 x \right )+18 x^{2} y-24 y \cos \relax (x ) x^{2}+x^{3} \cos \left (3 x \right )+15 x^{3} \cos \relax (x )-6 x^{3} \cos \left (2 x \right )-10 x^{3}}{4 x} \]

\[ {}y^{\prime } = -\frac {8 x \left (a -1\right ) \left (a +1\right )}{8+4 a^{4} y^{2} x^{2}+4 x^{2} y^{2}-4 a^{2} x^{6}-6 y^{4} a^{2} x^{2}-8 a^{2}+3 x^{2} y^{4}+2 x^{4}-6 a^{2} x^{4}-2 a^{2} y^{4}-2 a^{6} x^{4}+6 a^{4} x^{4}+3 x^{4} y^{2}-8 y+2 y^{4}-8 y^{2} a^{2} x^{2}-3 a^{6} y^{2} x^{4}+9 y^{2} a^{4} x^{4}-9 y^{2} a^{2} x^{4}+3 a^{4} y^{4} x^{2}-a^{2} y^{6}+a^{8} x^{6}-4 a^{6} x^{6}+6 a^{4} x^{6}+x^{6}+y^{6}} \]

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

\[ {}y^{\prime } = -\frac {1296 y}{216-648 x^{2} y^{2}+72 y^{8} x +216 y^{7} x -1728 y^{3}-648 x^{2} y-216 x^{2} y^{4}+216 x^{3}+1080 x y^{3}-315 y^{9}+216 x y^{2}-2376 y^{2}-432 x y+216 x^{2}-1296 y-1944 y^{4}-570 y^{8}+1152 x y^{4}+1080 y^{5} x -882 y^{6}-126 y^{10}-8 y^{12}-36 y^{11}+594 x y^{6}-324 x^{2} y^{3}-612 y^{5}-846 y^{7}} \]

\[ {}y^{\prime } = -\frac {x \left (-513-432 x -1296 x^{2} y^{2}-96 x^{8}-216 y^{3}-594 x^{2} y-756 x^{3}-576 x^{5}-864 x^{4}-144 x^{7}-540 y^{2}-972 x^{4} y^{2}-1134 x^{2}-288 y x^{6}+720 x^{3} y-378 y+864 y^{2} x^{5}+1008 x^{5} y-216 y x^{4}+64 x^{9}-648 y^{3} x^{4}-216 x^{6} y^{3}+432 y^{2} x^{7}-456 x^{6}+432 x^{3} y^{2}-216 y^{2} x^{6}-288 y x^{8}+288 y x^{7}-648 x^{2} y^{3}\right )}{216 \left (x^{2}+1\right )^{4}} \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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