2.14.22 Liouville \(u_{xx} + u_{yy} +e^{\lambda u} = 0\)

problem number 124

Added December 27, 2018.

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

Liouville equation. Solve for \(u(x,y)\) \[ u_{xx} + u_{yy} +e^{\lambda u} = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  Laplacian[u[x, y], {x, y}] + Exp[lam*u[x, y]] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

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

sol=()

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