Given the one-dimensional wave equation for a string stretched between two horizontal points at x = 0 and x = L as \frac{\partial^{2} u}{\partial x^{2}}=\frac{1}{C^{2}} \frac{\partial^{2} u}{\partial t^{2}}

where u(x, t) is the vertical displacement of the string, t is time and C is constant? The solution for the u (x, t) is u(x, t)=\sum_{n=1}^{\infty} \sin \left(P_{n} x\right)\left[D_{n} \cos \left(C P_{n} t\right)+E_{n} \sin \left(C P_{n} t\right)\right] P_{n}=\frac{n \pi}{L} \quad n=1,2,3, \ldots D_{n}=\frac{2}{L} \int_{0}^{L} f(x) \sin \left(P_{n} x\right) d x E_{n}=\frac{}{n Q \pi} \int_{0}^{L} g(x) \sin \left(P_{n} x\right) d x where f(x) is the initial displacement and g(x) is the initial velocity for 0 ≤ x ≤ L. With below boundary and initial conditions, solve the given wave equation. u(0, t)=0 \text { and } u(2, t)=0 \text { for } t \geq 0 \overline{\partial t}(x, 0)=0 \quad \text { for } 0 \leq x \leq 2 u(x, 0)=\left\{\begin{array}{ll}

-2 x & 0 \leq x \leq 1 \\

2 x-4 & 1 \leq x \leq 2

\end{array} \quad \text { for } 0 \leq x \leq 2\right.