### Instruction: Consider the following boundary value problem: 

$$
\frac{d^2 y}{dx^2} + 4y = 0, \quad y(0) = 0, \quad y(\pi) = 1.
$$

Use the Fourier transform to find the solution to this differential equation.### Response: To solve the given boundary value problem using the Fourier transform, we first need to rewrite the problem in terms of a Fourier series. Since the given boundary conditions are not homogeneous, we will first find a particular solution and then solve the homogeneous problem.

Let's find a particular solution $y_p(x)$ that satisfies the boundary conditions:

$$
y_p(0) = 0, \quad y_p(\pi) = 1.
$$

A simple choice for $y_p(x)$ is a linear function:

$$
y_p(x) = \frac{x}{\pi}.
$$

Now, let's consider the homogeneous problem:

$$
\frac{d^2 y_h}{dx^2} + 4y_h = 0, \quad y_h(0) = 0, \quad y_h(\pi) = 0.
$$

The general solution to the homogeneous problem is:

$$
y_h(x) = A\sin(2x) + B\cos(2x).
$$

Applying the boundary conditions:

$$
y_h(0) = A\sin(0) + B\cos(0) = B = 0,
$$

and

$$
y_h(\pi) = A\sin(2\pi) = 0.
$$

Since $A\sin(2\pi) = 0$ for any value of $A$, we can write the homogeneous solution as:

$$
y_h(x) = A\sin(2x).
$$

Now, the general solution to the original problem is the sum of the particular and homogeneous solutions:

$$
y(x) = y_p(x) + y_h(x) = \frac{x}{\pi} + A\sin(2x).
$$

However, we have already satisfied the boundary conditions with the particular solution, so we don't need to find the value of $A$. Thus, the solution to the given boundary value problem is:

$$
y(x) = \frac{x}{\pi} + A\sin(2x).
$$