Added June 3, 2019.
Problem 3.14(f) nonlinear pde’s by Lokenath Debnath, 3rd edition.
Solve for \(u(x,y)\) \[ y^2 u_x+(x y) u_y=x \] With \(u(x,1)=x^2\).
Mathematica ✓
ClearAll["Global`*"]; pde = y^2*D[u[x, y], x] +(x*y)*D[u[x, y], y]== x; ic=u[x,1]==x^2; sol = AbsoluteTiming[TimeConstrained[DSolve[{pde,ic} ,u[x, y], {x, y}], 60*10]];
\[\left \{\left \{u(x,y)\to x^2-y^2+\frac {\log \left (y^2\right )}{2}+1\right \}\right \}\]
Maple ✓
restart; pde :=y^2*diff(u(x,y),x) + (x*y)*diff(u(x,y),y)= x; ic := u(x,1)=x^2; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde,ic],u(x,y)) ),output='realtime'));
\[u \left ( x,y \right ) ={\frac {\ln \left ( {y}^{2} \right ) }{2}}+{x}^{2}-{y}^{2}+1\]
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