2.169   ODE No. 169

\[ (a x+b)^2 y'(x)+y(x)^3 (a x+b)+c y(x)^2=0 \] Mathematica : cpu = 2.11208 (sec), leaf count = 149


\[\text {Solve}\left [-\frac {c}{\sqrt {-a (a x+b)^2}}=\frac {2 \exp \left (\frac {1}{2} \left (-\frac {c}{\sqrt {-a (a x+b)^2}}-\frac {\left (-a (a x+b)^2\right )^{3/2}}{a y(x) (a x+b)^3}\right )^2\right )}{\sqrt {2 \pi } \text {erfi}\left (\frac {-\frac {c}{\sqrt {-a (a x+b)^2}}-\frac {\left (-a (a x+b)^2\right )^{3/2}}{a y(x) (a x+b)^3}}{\sqrt {2}}\right )+2 c_1},y(x)\right ]\] Maple : cpu = 0.136 (sec), leaf count = 153


\[\frac {\left (\sqrt {\pi }\, \sqrt {2}\, \erf \left (\frac {\sqrt {2}\, \left (c y \relax (x )+a \left (a x +b \right )\right )}{2 \sqrt {a}\, y \relax (x ) \left (a x +b \right )}\right ) {\mathrm e}^{\frac {\left (c y \relax (x )+a \left (a x +b \right )\right )^{2}}{2 y \relax (x )^{2} \left (a x +b \right )^{2} a}} a c +2 \left (a x +b \right ) a^{\frac {3}{2}}\right ) {\mathrm e}^{-\frac {\left (\left (-a x -b +c \right ) y \relax (x )+a \left (a x +b \right )\right ) \left (\left (a x +b +c \right ) y \relax (x )+a \left (a x +b \right )\right )}{2 y \relax (x )^{2} \left (a x +b \right )^{2} a}}+2 c_{1} a^{\frac {5}{2}}}{2 a^{\frac {5}{2}}} = 0\]