2.41   ODE No. 41

\[ a x y(x)^3+b y(x)^2+y'(x)=0 \] Mathematica : cpu = 0.251328 (sec), leaf count = 103


\[\text {Solve}\left [-\frac {b^2 \left (\frac {2 \tan ^{-1}\left (\frac {-2 a x y(x)-b}{b \sqrt {-\frac {4 a}{b^2}-1}}\right )}{\sqrt {-\frac {4 a}{b^2}-1}}-\log \left (\frac {a (-x) y(x) (-a x y(x)-b)-a}{a^2 x^2 y(x)^2}\right )\right )}{2 a}=-\frac {b^2 \log (x)}{a}+c_1,y(x)\right ]\] Maple : cpu = 0.304 (sec), leaf count = 103


\[y \relax (x ) = \frac {{\mathrm e}^{\RootOf \left (2 \sqrt {b^{2}+4 a}\, b \arctanh \left (\frac {2 a \,{\mathrm e}^{\textit {\_Z}}+b}{\sqrt {b^{2}+4 a}}\right )-\ln \left (x^{2} \left (a \,{\mathrm e}^{2 \textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}-1\right )\right ) b^{2}+2 c_{1} b^{2}+2 \textit {\_Z} \,b^{2}-4 \ln \left (x^{2} \left (a \,{\mathrm e}^{2 \textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}-1\right )\right ) a +8 c_{1} a +8 \textit {\_Z} a \right )}}{x}\]