Added June 3, 2019.
Problem 3.9(a) nonlinear pde’s by Lokenath Debnath, 3rd edition.
Solve for \(u(x,y)\) \[ u_x+x u_y=y \] With \(u(0,y)=y^2\).
Mathematica ✓
ClearAll["Global`*"]; pde = D[u[x, y], x] +x*D[u[x, y], y]== y; ic = u[0,y]==y^2; sol = AbsoluteTiming[TimeConstrained[DSolve[{pde,ic} ,u[x, y], {x, y}], 60*10]];
\[\left \{\left \{u(x,y)\to -x^2 y+\frac {x^4}{4}-\frac {x^3}{3}+x y+y^2\right \}\right \}\]
Maple ✓
restart; pde := diff(u(x,y),x) + x*diff(u(x,y),y)= y; ic := u(0,y)=y^2; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde,ic],u(x,y))),output='realtime'));
\[u \left ( x,y \right ) =-{\frac {{x}^{3}}{3}}+yx+ \left ( -{\frac {{x}^{2}}{2}}+y \right ) ^{2}\]
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