Added June 3, 2019.
Problem 3.17(c) nonlinear pde’s by Lokenath Debnath, 3rd edition.
Solve for \(u(x,y)\) \[ (x+y) u_x + (x-y)u_y =0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = (x+y)*D[u[x, y], x] +(x-y)*D[u[x, y], y]== 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde ,u[x, y], {x, y}], 60*10]];
\begin {align*} & \left \{u(x,y)\to c_1\left (\log \left (-\sqrt {-x^2+2 x y+y^2}\right )\right )\right \}\\& \left \{u(x,y)\to c_1\left (\frac {1}{2} \log \left (-x^2+2 x y+y^2\right )\right )\right \}\\ \end {align*}
Maple ✓
restart; pde :=(x+y)*diff(u(x,y),x) +(x-y)*diff(u(x,y),y)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y)) ),output='realtime'));
\[u \left (x , y\right ) = \textit {\_F1} \left (-\frac {1}{\sqrt {-x^{2}+2 x y +y^{2}}}\right )\]
____________________________________________________________________________________