2.667   ODE No. 667

\[ y'(x)=\frac {e^{-2 b x} y(x)^3}{e^{-b x} y(x)+1} \] Mathematica : cpu = 1.00245 (sec), leaf count = 90


\[\text {Solve}\left [\frac {\log (y(x))}{b}+\frac {1}{2} \left (-\frac {\log \left (y(x)^2-b e^{b x} \left (e^{b x}+y(x)\right )\right )}{b}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\frac {b}{b+4}} \left (2 e^{b x}+y(x)\right )}{y(x)}\right )}{\sqrt {b} \sqrt {b+4}}+2 x\right )=c_1,y(x)\right ]\] Maple : cpu = 0.283 (sec), leaf count = 82


\[b x -\frac {b \arctanh \left (\frac {-2 y \relax (x ) {\mathrm e}^{-b x}+b}{\sqrt {b^{2}+4 b}}\right )}{\sqrt {b^{2}+4 b}}-\frac {\ln \left (-b y \relax (x ) {\mathrm e}^{-b x}+y \relax (x )^{2} {\mathrm e}^{-2 b x}-b \right )}{2}+\ln \left (y \relax (x ) {\mathrm e}^{-b x}\right )-c_{1} = 0\]