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