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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.

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When the initial data consists of linear functions (or polynomials of low degree) one can sometimes find the solution by educated guess, possible involving some undetermined coefficients.

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.

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