Help with Method of Separation of Variables

You fail to mention how much the string is initially displaced by at $(x,t) = \left( \frac{L}{2} , 0 \right)$ . So let us call that value $M$ .

The partial differencial equation is the 1d wave equation:
$\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial u^2}{\partial x^2}$ .

The initial value problem is:
$\left\{ \begin{array}{c}u(x,0) = f(x) \\ u_t(x,0) = 0 \end{array}\right. \mbox{ for }0<x<L$ .

Where $f(x)$ is the function described in your curve, which is,
$f(x) = \left\{ \begin{array}{c} \frac{2M}{L}x \mbox{ for }0 < x \leq \frac{L}{2} \\ \frac{2M}{L}\left( L - x \right) \mbox{ for }\frac{L}{2} \leq x < L \end{array} \right.$ .

And the boundary value problem is:
$u(0,t) = u(L,t) = 0 \mbox{ for }t\geq 0$ .

See if you can solve this wave equation from heir.