# Solving the 3d initial value wave equation with linear data

Solve the following wave equation in three dimensions

$$u_{tt} - c^2\Delta u = 0,\quad t > 0$$ $$u(x, y, z, 0) = 0,\quad u_t(x, y, z, 0) = y$$

I tried to solve this example and I just got integral solution from which I cannot go further. I got stuck solving integral of specified value.

Here, looking at the initial values brings to mind $u(x,y,z,t)= ty$ which satisfies both of them. And because $y$ is a linear function, its Laplacian is zero... from where you can conclude that this is in fact the solution.