Added June 3, 2019.
Problem 3.3(g) nonlinear pde’s by Lokenath Debnath, 3rd edition.
Solve for \(u(x,y)\) \[ y^2 u_x- x y u_y=x(u-2 y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = y^2*D[u[x, y], x] - x*y*D[u[x, y], y] ==x*(u[x,y]-2*y); sol = AbsoluteTiming[TimeConstrained[DSolve[pde ,u[x, y], {x, y}], 60*10]];
\begin {align*} & \left \{u(x,y)\to \fbox {$\frac {\sqrt {y^2} c_1\left (\frac {1}{2} \left (x^2+y^2\right )\right )-x^2 \sqrt {-y^2}}{\sqrt {-y^4}}\text { if }x=0\lor y\geq 0$}\right \}\\& \left \{u(x,y)\to \fbox {$\frac {\sqrt {-y^2} x^2+\sqrt {y^2} c_1\left (\frac {1}{2} \left (x^2+y^2\right )\right )}{\sqrt {-y^4}}\text { if }x=0\lor y\leq 0$}\right \}\\ \end {align*}
Maple ✓
restart; pde :=y^2*diff(u(x,y),x)- x*y*diff(u(x,y),y)=x*(u(x,y)-2*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y))),output='realtime'));
\[u \left (x , y\right ) = -\frac {x^{2}}{y}+\frac {\textit {\_F1} \left (x^{2}+y^{2}\right )}{\sqrt {-y^{2}}}\]
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