Added June 3, 2019.
Problem 3.14(e) nonlinear pde’s by Lokenath Debnath, 3rd edition.
Solve for \(u(x,y)\) \[ x u_x+y u_y=x^2+y^2 \] With \(u(x,1)=x^2\) with \(x>0,y>0\).
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[u[x, y], x] +y*D[u[x, y], y]== x^2+y^2; ic=u[x,1]==x^2; sol = AbsoluteTiming[TimeConstrained[DSolve[{pde,ic} ,u[x, y], {x, y},Assumptions->{x>0,y>0}], 60*10]];
\[\left \{\left \{u(x,y)\to \frac {x^2 y^2+x^2+y^4-y^2}{2 y^2}\right \}\right \}\]
Maple ✓
restart; pde :=x*diff(u(x,y),x) + y*diff(u(x,y),y)= x^2+y^2; ic := u(x,1)=x^2; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde,ic],u(x,y)) assuming x>0,y>0),output='realtime'));
\[u \left ( x,y \right ) ={\frac {{x}^{2}}{2}}+{\frac {{y}^{2}}{2}}+{\frac {{x}^{2}}{2\,{y}^{2}}}-{\frac {1}{2}}\]
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