2.563   ODE No. 563

\[ a y(x)+b+x y'(x)+\log \left (y'(x)\right )=0 \] Mathematica : cpu = 0.20711 (sec), leaf count = 59


\[\text {Solve}\left [a \left (\frac {(a+1) \log \left (1-a W\left (x e^{-a y(x)-b}\right )\right )}{a^2}+\frac {W\left (x e^{-a y(x)-b}\right )}{a}\right )+a y(x)=c_1,y(x)\right ]\] Maple : cpu = 0.239 (sec), leaf count = 66


\[-\left ({\mathrm e}^{-a y \relax (x )-\LambertW \left (x \,{\mathrm e}^{-a y \relax (x )-b}\right )-b}\right )^{-\frac {1}{a +1}} c_{1}+x -\frac {{\mathrm e}^{a y \relax (x )+\LambertW \left (x \,{\mathrm e}^{-a y \relax (x )-b}\right )+b}}{a} = 0\]