Find the free-vibration solution of a cord fixed at both ends when its initial conditions are given by

\[w(x, 0)=0, \quad \frac{\partial w}{\partial t}(x, 0)=\frac{2 a x}{l} \quad \text { for } \quad 0 \leq x \leq \frac{l}{2}\]

and

\[\frac{\partial w}{\partial t}(x, 0)=2 a\left(1-\frac{x}{l}\right) \quad \text { for } \quad \frac{l}{2} \leq x \leq 1\]