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


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

Maple


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

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