Added June 3, 2019.
Problem 3.5(f) nonlinear pde’s by Lokenath Debnath, 3rd edition.
Solve for \(u(x,y)\) \[ u_x+x u_y=(y-\frac {x^2}{2})^2 \] with \(u(0,y)=e^{y}\).
Mathematica ✓
ClearAll["Global`*"]; pde = D[u[x, y], x] + x*D[u[x,y],y] == (y-x^2/2)^2; ic = u[0,y]==Exp[y]; sol = AbsoluteTiming[TimeConstrained[DSolve[{pde,ic} ,u[x, y], {x, y}], 60*10]];
\[\left \{\left \{u(x,y)\to \frac {1}{4} x \left (x^2-2 y\right )^2+e^{y-\frac {x^2}{2}}\right \}\right \}\]
Maple ✓
restart; pde := diff(u(x,y),x)+ x*diff(u(x,y),y)= (y-x^2/2)^2; ic := u(0,y)=exp(y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde,ic],u(x,y)) ),output='realtime'));
\[u \left ( x,y \right ) ={\frac {{x}^{5}}{4}}-y{x}^{3}+{y}^{2}x+{{\rm e}^{-{\frac {{x}^{2}}{2}}+y}}\]
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