Added June 3, 2019.
Problem 3.5(g) nonlinear pde’s by Lokenath Debnath, 3rd edition.
Solve for \(u(x,y)\) \[ x u_x+y u_y=u+1 \] with \(u=x^2\) on \(y=x^2\).
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[u[x, y], x] + y*D[u[x,y],y] == u[x,y]+1; ic = u[x,x^2]==x^2; sol = AbsoluteTiming[TimeConstrained[DSolve[{pde,ic} ,u[x, y], {x, y}], 60*10]];
\[\left \{\left \{u(x,y)\to \frac {x^2}{y}+y-1\right \}\right \}\]
Maple ✓
restart; pde :=x*diff(u(x,y),x)+ y*diff(u(x,y),y)= u(x,y)+1; ic := u(x,x^2)=x^2; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y)) ),output='realtime'));
\[u \left ( x,y \right ) ={\it \_F1} \left ( {\frac {y}{x}} \right ) x-1\] Maple does not accept this form of Cauchy data as I.C.
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