2.667   ODE No. 667

\[ y'(x)=\frac {e^{-2 b x} y(x)^3}{e^{-b x} y(x)+1} \]

✓ Mathematica : cpu = 2.84321 (sec), leaf count = 95

DSolve[Derivative[1][y][x] == y[x]^3/(E^(2*b*x)*(1 + y[x]/E^(b*x))),y[x],x]
 

\[\text {Solve}\left [\frac {\log (y(x))}{b}+\frac {-\log \left (b e^{b x} \left (e^{b x}+y(x)\right )-y(x)^2\right )+2 \sqrt {\frac {b}{b+4}} \tanh ^{-1}\left (\frac {\sqrt {\frac {b}{b+4}} \left (2 e^{b x}+y(x)\right )}{y(x)}\right )+2 \log \left (e^{b x}\right )}{2 b}=c_1,y(x)\right ]\]

✓ Maple : cpu = 0.257 (sec), leaf count = 82

dsolve(diff(y(x),x) = y(x)^3/(y(x)*exp(-b*x)+1)*exp(-2*b*x),y(x))
 

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