"Solve the wave equation: \begin{cases} u_{tt}(x,t)=c^2u_{xx}(x,t), 0<x<\pi, t>0 \\ u(0,t)=t, u(\pi,t)=(1+\pi)t,\\ u(x,0)=0,\\ u_{t}(x,0)=\sin(x)+x+1 \end{cases} Hint: Consider $u_s(x,t)$ a linear function in $x$, such that $u_s(0,t)=t$, and $u_s(\pi,t)=(1+\pi)t$. Solve a new problem for $v(x,t)=u(x,t)-u_s(x,t)$."
Following the hint, I have $v(x,t)$ homogeneous wave equation with homog BCs. I suppose I can solve that but how do I deduce $u(x,t)$ from that?
I tried to find problems like this on the net so I can get a feel of how to solve wave equation with inhomo BCs but I had no luck.
Substitute $\,u(x,t)=v(x,t)+t+xt\,$ and consider the problem $$ \begin{cases} v_{tt}=c^2v_{xx}, \quad 0<x<\pi,\;t>0;\\ v(0,t)=v(\pi,t),\quad t\geqslant 0;\\ v(x,0)=0,\;\;v_t(x,0)=\sin{x},\quad 0\leqslant x\leqslant \pi. \end{cases} $$ Consider the Sturm–Liouville eigenvalue problem $$ X''=\lambda X, \quad 0<x<\pi,\\ X(0)=X(\pi)=0, $$ and find its solution $$ X_n=\sin{(nx)},\;\;\lambda_n=-n^2,\;\; n\geqslant 1. $$ Solution $\,v=v(x,t)\,$ is constructed as Fourier series $$ v(x,t)=\sum_{n=1}^{\infty}T_n(t)\sin{(nx)} $$ with coefficients $\,T_n=T_n(t)\,$ to be found by solving the Cauchy problems $$ T''_n=-c^2n^2T_n\,,\;\;t>0;\quad T_n(0)=0,\;\; T'_n(0)=\begin{cases} 1,\quad n=1,\\ 0,\quad n\neq 1. \end{cases} $$ It is clear that $$ T_n(t)=\begin{cases} \frac{1}{c}\sin{(ct)},\quad n=1,\\ 0,\quad n\neq 1. \end{cases} $$ Hence $\,v(x,t)=T_1(t)\!\cdot\!\sin{x}=\frac{1}{c}\sin{(ct)}\!\cdot\!\sin{x}$, which results in $u(x,t)=t+xt +\frac{1}{c}\sin{(ct)}\!\cdot\!\sin{x}$.