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