Solving Damped Wave Equation with Initial Conditions

[Dustinsfl]

$$
u_{tt} + 3u_t = u_{xx}
$$
$$
u(0,t) = u(\pi,t) = 0
$$
$$
u(x,0) = 0\quad\text{and}\quad u_t(x,0) = 10.
$$
\begin{alignat*}{3}
u(x,t) & = & \exp\left[-\frac{3t}{2}\right]\sin x\left[A_1\cosh\frac{t\sqrt{5}}{2} + B_1\sinh\frac{t\sqrt{5}}{2}\right]\\
& + & \exp\left[-\frac{3t}{2}\right]\sum_{n = 2}^{\infty}\sin nx\left[C_n\cos \left(t\frac{\sqrt{4n^2 - 9}}{2}\right) + D_n\sin\left(t\frac{\sqrt{4n^2 - 9}}{2}\right)\right]
\end{alignat*}
The hyperbolic part is when n = 1 which would be overdamped and the rest are the underdamped modes.
I have solved for all the coefficients except ##B_1##.
##A_1 = C_n = 0## and ##D_n = \begin{cases} 0, & \text{of n is even}\\ \frac{80}{n\pi\sqrt{4n^2 - 9}}, & \text{if n is odd}\end{cases}##
However, I haven't been able to solve for ##B_1##. Help would be much appreciated.
\begin{alignat*}{3}
u(x,t) & = & B_1\exp\left[-\frac{3t}{2}\right]\sin x \sinh\frac{t\sqrt{5}}{2}+ \exp\left[-\frac{3t}{2}\right]\sum_{n = 2}^{\infty}D_n\sin nx\sin\left(t\frac{\sqrt{4n^2 - 9}}{2}\right)
\end{alignat*}

 

[lippyka]

Since ##u(x,0) = 0##, $$B_1\sin x \sinh\frac{t\sqrt{5}}{2}+ \sum_{n = 2}^{\infty}D_n\sin nx\sin\left(t\frac{\sqrt{4n^2 - 9}}{2}\right) = 0$$Since the second part on the left side is zero for ##t = 0##, $$B_1\sin x \sinh\frac{t\sqrt{5}}{2} = 0$$Since ##\sin x## is never zero, $$B_1\sinh\frac{t\sqrt{5}}{2} = 0$$Therefore, $$B_1 = 0$$