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I'm going through my notes and I'm stuck with quite a basic part of the wave equation.

I have that $u_{tt}=c^2u_{xx}$ for $ x ∈ (0,1), t>1$

subject to the following boundary conditions:

$u(0, t) = u(1, t) = 0 , t > 0$

I am also told that the initial displacement is:

$u(x, 0) = f(x), x ∈ (0, 1)$

and also that it is initially at rest which I believe means that

$ u_{t}(x, 0) = g(x) = 0 $

All I needed to do from here was find an expression for $h_1(z) $ and $h_2(z) $ for $0 ≤ z ≤ 1$

When deriving D'Alembert's formula I noticed that:

$h_1(z)+h_2(z)=f(z)$ and $-h_1(z)+h_2(z)=\frac{1}{c}\int_{0}^{z}g(s)ds-\frac{1}{2}A$

And from this I deduced that $h_1(z)=h_2(z)=\frac{1}{2}f(z)$

After this I need to substitute in the boundary condition at x = 0 into the general solution derive an expression for $h_1(z)$ in terms of $h_2$ for z < 0, and then also substitute in the boundary condition at x = 1 into the general solution derive an expression for $h_2(z)$ in terms of $h_1$ for z > 1

But this is where I get stuck, I know the answer to the first bit is $h_1(z)=−h_2(−z)$ but I'm just not sure how to get there.

Any help would be greatly appreciated.

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