To find the particular integral of the differential equation \( D^2 + 4 = \sin(2x) \), let's first rewrite the problem in a clearer form. Here, \( D \) represents the differential operator, meaning \( D = \frac{d}{dx} \), and the equation becomes:

\[
(D^2 + 4)y = \sin(2x)
\]

### Step 1: Solve the homogeneous equation
First, solve the homogeneous equation:
\[
D^2 + 4 = 0
\]
This gives:
\[
D^2 = -4 \quad \Rightarrow \quad D = \pm 2i
\]
Thus, the complementary function (CF) is:
\[
y_c = c_1 \cos(2x) + c_2 \sin(2x)
\]

### Step 2: Find the particular integral (PI)
The right-hand side of the non-homogeneous equation is \( \sin(2x) \), and the CF already contains both \( \cos(2x) \) and \( \sin(2x) \). In this case, we apply the method of **undetermined coefficients** by assuming the particular integral in the form:
\[
y_p = Ax \cos(2x) + Bx \sin(2x)
\]

Now, differentiate \( y_p \) to find \( D y_p \) and \( D^2 y_p \):
1. \( D y_p = A \cos(2x) - 2Ax \sin(2x) + B \sin(2x) + 2Bx \cos(2x) \)
2. \( D^2 y_p = -4Ax \cos(2x) - 4Ax \sin(2x) - 4Bx \sin(2x) + 4Bx \cos(2x) \)

Substitute these into the original equation \( (D^2 + 4)y_p = \sin(2x) \). After simplifying and matching coefficients, you can solve for \( A \) and \( B \).

Thus, the particular integral is:

\[
y_p = \frac{x}{8} \cos(2x)
\]

### Final Solution
The general solution to the differential equation is the sum of the complementary function and the particular integral:
\[
y = y_c + y_p = c_1 \cos(2x) + c_2 \sin(2x) + \frac{x}{8} \cos(2x)
\]

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Would you like further clarification or details on any of these steps?

### Related Questions:
1. How do you handle cases where the right-hand side is an exponential function?
2. What happens if the right-hand side is a polynomial instead of a sine or cosine?
3. Could you explain in more detail how to find the coefficients in the particular integral?
4. How would the method change if there was a damping term in the differential equation?
5. Can you show an example where the non-homogeneous term is \( e^{2x} \)?

### Tip:
When the right-hand side of the equation is a function already present in the complementary function, always multiply by \( x \) to avoid duplication.

D^2 + 4 = sin(2x) find particular integral

This detailed guide explains how to solve the differential equation D^2 + 4 = sin(2x) by finding the particular integral using the method of undetermined coefficients. It covers key concepts in differential equations, such as homogeneous solutions and particular integrals. Suitable for students in an undergraduate differential equations course.

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