(ux + uy)2 − u2 = 0. Problem 3.10 Lokenath Debnath

2.1.60 $(u_{x} + u_{y})^{2} - u^{2} = 0$ . Problem 3.10 Lokenath Debnath

problem number 60

Added June 3, 2019.

Problem 3.10 nonlinear pde’s by Lokenath Debnath, 3rd edition.

Solve for $u (x, y)$ $(u_{x} + u_{y})^{2} - u^{2} = 0$

Mathematica ✓

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

$\begin{aligned} {u (x, y) \to e^{- x} c_{1} (y - x)} \\ {u (x, y) \to e^{x} c_{1} (y - x)} \end{aligned}$

Maple ✓

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

$u (x, y) = c_{1} e^{\frac{y {_c}_{2} + x}{{_c}_{2} + 1}}$

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2.1.60 (ux+uy)2−u2=0. Problem 3.10 Lokenath Debnath

2.1.60 $(u_{x} + u_{y})^{2} - u^{2} = 0$ . Problem 3.10 Lokenath Debnath