A string is tightly stretched, and its ends are fastened at two points x = 0 and x = L. Initially, the string is in the form f(x) = kx(L - x), and then released from that position. Find the displacement at any point on the string at a distance x from one end and at any time t > 0

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)}

A string is tightly stretched, and its ends are fastened at two points x = 0 and x = L. Initially, the string is in the form f(x) = kx(L - x), and then released from that position. Find the displacement at any point on the string at a distance x from one end and at any time t > 0.

Explore the solution to the wave equation for a string fastened at both ends and released from the initial shape f(x) = kx(L - x). This solution uses separation of variables and is ideal for undergraduate students studying partial differential equations.

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