Added June 3, 2019.
Problem 3.14(g) nonlinear pde’s by Lokenath Debnath, 3rd edition.
Solve for \(u(x,y)\) \[ x u_x+y u_y=x y \] With \(u=\frac {x^2}{2}\) at \(y=x\).
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[u[x, y], x] +y*D[u[x, y], y]== x*y; ic=u[x,x]==x^2/2; sol = AbsoluteTiming[TimeConstrained[DSolve[pde ,u[x, y], {x, y}], 60*10]];
\[\left \{\left \{u(x,y)\to \frac {x y}{2}+c_1\left (\frac {y}{x}\right )\right \}\right \}\]
Maple ✓
restart; pde :=x*diff(u(x,y),x) + y*diff(u(x,y),y)= x*y; ic := u(x,x)=x^2/2; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y)) ),output='realtime'));
\[u \left ( x,y \right ) ={\frac {yx}{2}}+{\it \_F1} \left ( {\frac {y}{x}} \right ) \]
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