(Solved): Wave Equation problem...   I solved it but i am stuck at the final step     ...

Wave Equation problem...

 
I solved it but i am stuck at the final step

 

 

6. Solve the following partial D.E. : \[ \begin{array}{l} 4 \frac{\sigma^{2} U}{\sigma x^{2}}=\frac{\sigma^{2} U}{\sigma t^{2}} \quad U(0, t)=0, U(\pi, t)=0 \\ U(x, 0)=x(\pi-x), \frac{\sigma(t}{\sigma t}(x, 0)=00 \leq x \leq \pi \\ \text { (7 Mlarks) (K1 I2 P1) } \\ \text { GOOD LUCK } \end{array} \] \( \begin{array}{cl}4 U_{x x}=U_{t t} & * U(0, t)=0 \\ U_{x x}=G F^{\prime \prime}, U_{t t}=\ddot{G} F & * U(x, t)=0 \\ U(x, t)=F(x) \cdot G(t) & * U_{t}(x, 0)=0 \\ 4 G F^{\prime \prime}=\ddot{G} F \\ 4 \frac{F^{\prime \prime}}{F}=\frac{\dot{G}}{G}=-\mathbb{P}^{2}\end{array} \) \( \begin{array}{l} U(x, t)=\sum_{n=0}^{\infty} F \cdot G \\ =\sum_{n=0}^{\infty} C_{2} \sin \left(n p_{2}\right)\left[C_{3} \cos (4 n t)+C_{4} \sin (4 n t)\right] \\ =\sum_{n=0}^{\infty} A \sin \left(n-\cos (4 n t)+B \sin \left(n p_{x}\right) \sin (4 n t)\right. \\ U_{t}(0,0)=0 \\ U_{t}=\sum_{n=0}^{\infty} A \sin (n x)[-4 n \sin (4 n t)]+B \sin (n p x)[4 n \cos (4 n t)] \\ 0=\sum_{n=0}^{\infty} B \sin (n x)[4 n \cos (4 n t)] \rightarrow B=0 \\ U(x, t)=\sum_{n=0}^{\infty} A \sin (n x) \cos (4 n t) \\ U(x, 0)=x(\pi-x) \\ x(\pi-x)=\sum_{n=0}^{\infty} A \sin (n)\end{array} \)