$u_{tt} = 2(u_{xx}+ u_{yy}) $

Question

I was given this equation $$u_{tt} = 2(u_{xx}+ u_{yy}) $$ and asked which one of the following will be the solution.

$1)$ $u(x , y , t) = t \sin (x+y^2)$

$2)$ $u(x , y , t) = t \cos (x+y^2) \sin (x+y^2)$

$3)$ $u(x , y , t) = \cos x \cos y \cos 2t$

I saw $u(x , y , t) = \cos x \cos y \cos 2t$ is satisfying the given partial differential equation and told him that this is the solution.

Have I gone wrong anywhere ?

Answer 1

No, you are correct. When you plug $u(x,y,t) = \cos x \cos y \cos (2t)$ into the equation, you see it is satisfied. So, this is a solution. It is not the solution because no initial or boundary conditions are specified (there are other solutions).