Determine the free$-$vibration solution of a string fixed at both ends under the initial conditions $w(x,0)=w_{0}sin(\frac{x}{l})$ and $\frac{\text{d}\text{e}\text{l}\text{w}}{\text{d}\text{e}\text{l}\text{t}}(x,0)=0.$

Hint: The free vibration solution for a string can is given by $w(x,t)={\displaystyle \underset{n=1}{\overset{}{}}}sin(\frac{nx}{l})(C_{n}cos(\frac{nct}{l})+D_{n}sin(\frac{nct}{l})),$ where $C_n$ and $D_n$ are Fourier coefficients given by $C_n=\frac{2}{l}{\displaystyle \underset{0}{\overset{l}{}}}{w}_0(x)sin(\frac{nx}{l}dx)$ and $D_n=\frac{2}{nc}{\displaystyle \underset{0}{\overset{l}{}}}{w}_0^{}(x)sin(\frac{nx}{l}dx).$