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.