6.1.6 problem number 6

problem number 422

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'));
 

\[w \left ( x,y \right ) = \left ( \int \!g \left ( x,y \right ) {{\rm e}^{-\int \!f \left ( x,y \right ) \,{\rm d}y}}\,{\rm d}y+{\it \_F1} \left ( x \right ) \right ) {{\rm e}^{\int \!f \left ( x,y \right ) \,{\rm d}y}}\]