2.563   ODE No. 563

\[ a y(x)+b+x y'(x)+\log \left (y'(x)\right )=0 \]

✓ Mathematica : cpu = 0.13645 (sec), leaf count = 59

DSolve[b + Log[Derivative[1][y][x]] + a*y[x] + x*Derivative[1][y][x] == 0,y[x],x]
 

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

✓ Maple : cpu = 0.25 (sec), leaf count = 66

dsolve(ln(diff(y(x),x))+x*diff(y(x),x)+a*y(x)+b=0,y(x))
 

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