2.15.23 Plateau \((1+u_y^2)u_{xx} - 2 u_x u_y y_{xy} + (1+u_x^2) u_{yy} = 0\)

problem number 133

Added December 27, 2018.

Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations

Plateau. Solve for \(u(x,y)\) \[ (1+u_y^2)u_{xx} - 2 u_x u_y y_{xy} + (1+u_x^2) u_{yy} = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  (1 + D[u[x, y], y]^2)*D[u[x, y], {x, 2}] - 2*D[u[x, y], x]*D[u[x, y], y]*D[u[x, y], x, y] + (1 + D[u[x, y], x]^2)*D[u[x, y], {y, 2}] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, y], {x, y}], 60*10]];
 

Failed

Maple

restart; 
pde :=(1+diff(u(x,y),y)^2)* diff(u(x,y),x$2)-2* diff(u(x,y),x)* 
       diff(u(x,y),y)*diff(u(x,y),x,y)+ 
       (1+diff(u(x,y),x)^2)*diff(u(x,y),y$2)=0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y))),output='realtime'));
 

\[u \left ( x,y \right ) ={\it \_C7}\, \left ( \tanh \left ( \left ( x-iy \right ) {\it \_C2}+{\it \_C1} \right ) \right ) ^{3}+{\it \_C5}\,\tanh \left ( \left ( x-iy \right ) {\it \_C2}+{\it \_C1} \right ) +{\it \_C4}\]

____________________________________________________________________________________