2.1.25 \(f_x + (f_y)^2 = f(x,y,z)+z\)

problem number 25

Taken from Maple pdsolve help pages, probem 5

Nonlinear first order PDE

Solve for \(f(x,y,z)\)

\begin{align*} f_x + (f_y)^2 &= f(x,y,z)+z \end{align*}

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[f[x, y, z], x] + D[f[x, y, z], y]^2 == f[x, y, z] + z; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, f[x, y, z], {x, y, z}], 60*10]];
 

\[\left \{\left \{f(x,y,z)\to \frac {1}{4} \left ((c_1(z)){}^2 W\left (-\frac {\exp \left (\frac {(x-1) c_1(z)+c_2(z)+y}{c_1(z)}\right )}{c_1(z)}\right ){}^2+2 (c_1(z)){}^2 W\left (-\frac {\exp \left (\frac {(x-1) c_1(z)+c_2(z)+y}{c_1(z)}\right )}{c_1(z)}\right )-4 z\right )\right \}\right \}\]

Maple ✓

restart; 
pde := diff(f(x,y,z),x) + (diff(f(x,y,z),y))^2 = f(x,y,z)+z; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,f(x,y,z),'build')),output='realtime'));
 

\[f \left (x , y , z\right ) = \frac {-{\mathrm e}^{2 x} c_{3}^{2}-c_{5} \left (\left (c_{3} y +c_{4} z +c_{1} \right ) {\mathrm e}^{x}+c_{5} z \right )}{c_{5}^{2}}\]

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