I have this problem: Find u(x,t) that satisfies:

$\partial_{xx}u(x,t)+\partial_{tt}u(x,t)=0$, for $x\in(0, \pi)$ and $t>0$.
$u(0,t) = 0 $, for t>0
$u(\pi,t) = 1 $, for t>0
$u(x,0) = \frac{x}{\pi} $, for $0 < x < \pi$

I have already solved the problem.

Assume u(x,t) = X(x) T(t)

From BC2:
X($\pi$) T(t) = 1
T(t) = $\frac{1}{X(\pi)}$, {constant function}

Our solution depends only on one variable
$\therefore u(x,t) = X(x) T(t)= \frac{x}{\pi}$

This is a solution to the problem, but as I found that this an an ill-posed problem which means this is not a unique solution.

My question is how to find an explicit solution to the given problem?


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