Added May 27, 2019.
From UMN Math 5587 HW2, Fall 2016, problem 5(a).
Solve for \(u(x,y)\) \begin {align*} u_x u_y = 1 \end {align*}
Mathematica ✓
ClearAll["Global`*"]; pde = D[u[x, y], x] * D[u[x, y], y] == 1; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, y], {x, y}], 60*10]];
\[\left \{\left \{u(x,y)\to \frac {x}{c_2}+c_2 y+c_1\right \}\right \}\]
Maple ✓
restart; pde := diff(u(x,y),x)*diff(u(x,y),y)=1; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y),HINT=`+`,'build')),output='realtime'));
\[u \left ( x,y \right ) ={\frac {x}{{\it \_c}_{{2}}}}+{\it \_C1}+{\it \_c}_{{2}}y+{\it \_C2}\]
Hand solution
Solve \(u_{x}u_{y}=1\). Let \(u=f\left ( x\right ) +g\left ( y\right ) \). The pde becomes \(f^{\prime }\left ( x\right ) g^{\prime }\left ( y\right ) =1\) or \(f^{\prime }\left ( x\right ) =\frac {1}{g^{\prime }\left ( y\right ) }\). Hence both are constant, say \(\lambda \). Therefore \(f^{\prime }\left ( x\right ) =\lambda \) or \(f\left ( x\right ) =\lambda x+C_{1}\) and similarly, \(\frac {1}{g^{\prime }\left ( y\right ) }=\lambda \) or \(g^{\prime }\left ( y\right ) =\frac {1}{\lambda }\) or \(g\left ( y\right ) =\frac {y}{\lambda }+C_{2}\). Therefore the solution becomes\begin {align*} u\left ( x,y\right ) & =f\left ( x\right ) +g\left ( y\right ) \\ & =\lambda x+C_{1}+\frac {y}{\lambda }+C_{2} \end {align*}
____________________________________________________________________________________