Consider the Boundary-Initial Value problem
$$9 \frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2}, \quad 0 < x < 3, \quad t > 0$$
$$u(0, t) = 0, \quad u(3, t) = 0, \quad t > 0$$
$$u(x, 0) = x(3 - x), \quad \frac{\partial u}{\partial t}(x, 0) = 0, \quad 0 < x < 3$$
This models the displacement $u(x, t)$ of a freely vibrating string, with fixed ends, initial profile $x(3 - x)$, and zero
initial velocity.
The solution $u(x, t)$, is given by the series
$$u(x, t) = \frac{4}{\pi^3} \sum_{n=1}^{\infty} b_n \sin\left(n \frac{\pi}{3} x\right) \cos(c_n t)$$
where
$$b_n = \text{?}$$
and
$$c_n = n\pi$$
