The problem involves finding the displacement $u(x, t)$ of a tightly stretched string, where the string is fixed at both ends $x = 0$ and $x = L$, and the initial shape of the string is given by $f(x) = kx(L - x)$. This is a standard wave equation problem, and we will solve it using the method of separation of variables.

Step 1: Wave Equation Setup

The wave equation for a string stretched between two points is given by:

$\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$

where $u(x, t)$ is the displacement of the string, and $c$ is the wave speed.

Step 2: Boundary Conditions

The string is fastened at the ends $x = 0$ and $x = L$, so the boundary conditions are:

$u(0, t) = 0 \quad \text{and} \quad u(L, t) = 0 \quad \text{for all } t$

Step 3: Initial Conditions

1. Initial displacement: At $t = 0$, the string is initially in the form $f(x) = kx(L - x)$, so: $u(x, 0) = kx(L - x)$

2. Initial velocity: The string is released from rest, so the initial velocity is zero: $\frac{\partial u}{\partial t}(x, 0) = 0$

Step 4: Solution Using Separation of Variables

To solve this, we use the method of separation of variables. We assume the solution takes the form:

$u(x, t) = X(x)T(t)$

Substituting this into the wave equation:

$X(x)T''(t) = c^2 X''(x)T(t)$

Dividing both sides by $X(x)T(t)$:

[ \frac{T''(t)}