Use D'Alembert's solution to solve the following

$\displaystyle u_{xx} - \frac{1}{c^2} u_{tt} = 4$

subject to the initial conditions $\displaystyle u(x, 0) = \sin (x)$ and $\displaystyle u_t (x, 0) = c \ \cos (x)$.

Note that $\displaystyle u_{xx} - c^{-2} u_{tt} = 4u_{\xi \eta}$ with $\displaystyle \xi = x + ct$ and $\displaystyle \eta = x - ct$