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The given image appears to show a third-order linear non-homogeneous differential equation of the form:

\[
\frac{d^3 y}{dx^3} + 2\frac{d^2 y}{dx^2} + \frac{dy}{dx} = e^{2x} + \sin(2x)
\]

To solve for the particular integral, we need to address the non-homogeneous part of the equation, which includes \( e^{2x} \) and \( \sin(2x) \). Here's the typical procedure to find the particular integral for such an equation:

### 1. Solve the Complementary Function (CF)
First, solve the homogeneous part of the equation:

\[
\frac{d^3 y}{dx^3} + 2\frac{d^2 y}{dx^2} + \frac{dy}{dx} = 0
\]

This will involve solving the characteristic equation, which is obtained by assuming \( y = e^{rx} \), substituting into the homogeneous equation, and solving for \( r \).

### 2. Particular Integral (PI) for \( e^{2x} \)
For the term \( e^{2x} \), assume a particular solution of the form:
\[
y_p = A e^{2x}
\]
where \( A \) is a constant to be determined. Substitute this into the left-hand side of the differential equation and solve for \( A \).

### 3. Particular Integral (PI) for \( \sin(2x) \)
For the \( \sin(2x) \) term, assume a particular solution of the form:
\[
y_p = B \sin(2x) + C \cos(2x)
\]
where \( B \) and \( C \) are constants to be determined. Substitute this into the differential equation and solve for \( B \) and \( C \).

Once you find the complementary function and the particular integral, the general solution will be:

\[
y = y_{\text{CF}} + y_{\text{PI}}
\]

Would you like to proceed with the detailed solution for the complementary function and particular integral, or focus on a specific part of this process?

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### Related Questions:
1. How do you derive the characteristic equation for a third-order differential equation?
2. What is the general method for solving non-homogeneous differential equations?
3. How do you handle terms like \( e^{2x} \) when solving differential equations?
4. What approach is used to solve for a particular integral with trigonometric terms?
5. Can you explain how to combine complementary and particular solutions?

#### Tip:
For higher-order differential equations, the superposition principle allows you to combine solutions corresponding to different non-homogeneous terms independently.

Find the particular integral of the given third-order linear non-homogeneous differential equation.

Learn how to solve a third-order linear non-homogeneous differential equation. This problem involves finding the particular integral for the terms e^(2x) and sin(2x) using the method of undetermined coefficients. Suitable for college-level calculus students.

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