Added January 2, 2019.
Problem 1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_y = w f(x,y)+ g(x,y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], y] == w[x, y]*f[x, y] + g[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^yf(x,K[1])dK[1]\right ) \left (\int _1^y\exp \left (-\int _1^{K[2]}f(x,K[1])dK[1]\right ) g(x,K[2])dK[2]+c_1(x)\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),y)=w(x,y)*f(x,y)+g(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int {\mathrm e}^{-\left (\int f \left (x , y\right )d y \right )} g \left (x , y\right )d y +\textit {\_F1} \left (x \right )\right ) {\mathrm e}^{\int f \left (x , y\right )d y}\]