Added June 3, 2019.
Problem 3.17(b) nonlinear pde’s by Lokenath Debnath, 3rd edition.
Solve for \(u(x,y)\) \[ x u(u^2+x y)u_x - y u(u^2+x y) u_y = x^4 \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*u[x,y]*(u[x,y]^2+x*y)*D[u[x, y], x] -y*u[x,y]*(u[x,y]^2+x*y)*D[u[x, y], y]== x^4; sol = AbsoluteTiming[TimeConstrained[DSolve[pde ,u[x, y], {x, y}], 60*10]];
\begin {align*} & \left \{u(x,y)\to -\sqrt {-x y-\sqrt {4 c_1(x y)+x^2 y^2+x^4}}\right \}\\& \left \{u(x,y)\to \sqrt {-x y-\sqrt {4 c_1(x y)+x^2 y^2+x^4}}\right \}\\& \left \{u(x,y)\to -\sqrt {-x y+\sqrt {4 c_1(x y)+x^2 y^2+x^4}}\right \}\\& \left \{u(x,y)\to \sqrt {-x y+\sqrt {4 c_1(x y)+x^2 y^2+x^4}}\right \}\\ \end {align*}
Maple ✓
restart; pde :=x*u(x,y)*(u(x,y)^2+x*y)*diff(u(x,y),x) -y*u(x,y)*(u(x,y)^2+x*y)*diff(u(x,y),y)=x^4; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y)) ),output='realtime'));
\[u \left ( x,y \right ) =\sqrt {-yx-\sqrt {{x}^{4}+{y}^{2}{x}^{2}+4\,{\it \_F1} \left ( yx \right ) }}\]
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