\begin{align} u_{tt} &= 4u_{xx}+(1-x)\,\cos t, \quad 0 <x < \pi, \quad t>0, \\[1em] u_x(0,t) &= \cos t - 1, ~\quad u_x(\pi,t) = \cos t, \qquad~~~~~~~~ t \geq 0, \\ u(x,0) &= \frac{x^2 \pi}{2} ~\quad~~~~~~ u_t(x,0) = \cos(3x), \quad 0<x<\pi \end{align}
Using $u(x,t)=w(x,t)+v(x,t)$ I can make the boundary conditions homogenous but I am stumped after that.