Added June 3, 2019.
Problem 3.8(d) nonlinear pde’s by Lokenath Debnath, 3rd edition.
Solve for \(u(x,y,z)\) \[ y z u_x - x z u_y+ x y (x^2+y^2) u_z=0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = y*z*D[u[x, y,z], x] - x*z*D[u[x, y,z], y]+x*y*(x^2+y^2)*D[u[x, y,z], z]== 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde ,u[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{u(x,y,z)\to c_1\left (\frac {1}{2} \left (x^2+y^2\right ),\frac {1}{2} \left (-x^2 y^2-x^4+z^2\right )\right )\right \}\right \}\]
Maple ✓
restart; pde :=y*z*diff(u(x,y,z),x) - x*z*diff(u(x,y,z),y)+x*y*(x^2+y^2)*diff(u(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y,z))),output='realtime'));
\[u \left ( x,y,z \right ) ={\it \_F1} \left ( {x}^{2}+{y}^{2},-{x}^{4}-{x}^{2}{y}^{2}+{z}^{2} \right ) \]
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