I am studying for an exam and am stumped by this because everything is non homogenous. The question is to solve for $u$,

\begin{align*} u_{tt} &= u_{xx} + q, 0 < x < 1, t>0\\ u(0,t) &= 0, t>0\\ u(1,t) &= sint, t>0\\ u(x,0) &= u_t(x,0) = 0, 0<x<1 \end{align*}

I have first tired to get rid of the non homogenous boundary condition $u(1,t) = sint$ by saying defining $w$ to be,

$w(x,t) = u(x,t) - xsint$

so the equation becomes,

\begin{align*} w_{tt} &= w_{xx} + q - xcost\\ w(0,1) &= 0\\ w(1,t) &= 0\\ w_t(x,0) &= -x\\ w(x,0) &= x(1-x)\\ \end{align*}

I am confused about how to proceed, can anyone help?


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