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The given differential equation is:

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

To find the **particular integral**, we can use the method of undetermined coefficients or the annihilator method. The right-hand side consists of two terms: \(e^{2x}\) and \(\sin(3x)\), so the particular integral will be found for each of these separately.

### Step-by-step solution:

1. **Solve for the particular integral of \(e^{2x}\)**:
   - The corresponding homogeneous equation is:
     \[
     D^3 y + 2D^2 y + D y = 0
     \]
     where \(D\) represents differentiation with respect to \(x\).
   - The right-hand side \(e^{2x}\) suggests that we try a solution of the form \(y_p = A e^{2x}\).
   - Substituting this into the left-hand side of the equation:
     \[
     (D^3 + 2D^2 + D)(A e^{2x}) = A(8 + 8 + 2)e^{2x} = 18A e^{2x}
     \]
     - Matching coefficients with the right-hand side \(e^{2x}\), we get:
       \[
       18A = 1 \implies A = \frac{1}{18}
       \]
     - Therefore, the particular integral for the \(e^{2x}\) term is:
       \[
       y_{p1} = \frac{1}{18}e^{2x}
       \]

2. **Solve for the particular integral of \(\sin(3x)\)**:
   - The term \(\sin(3x)\) suggests we try a solution of the form:
     \[
     y_{p2} = B \cos(3x) + C \sin(3x)
     \]
   - Substituting into the left-hand side:
     \[
     (D^3 + 2D^2 + D)(B \cos(3x) + C \sin(3x)) = (-27B \cos(3x) - 27C \sin(3x)) + (-18B \cos(3x) - 18C \sin(3x)) + (-3B \sin(3x) + 3C \cos(3x))
     \]
     Grouping the terms for \(\cos(3x)\) and \(\sin(3x)\), we get:
     \[
     \cos(3x) : (-27B - 18B + 3C) = 0 \quad \text{and} \quad \sin(3x) : (-27C - 18C - 3B) = 1
     \]
     Solving these two equations:
     - \(-45B + 3C = 0 \implies C = 15B\)
     - \(-45C - 3B = 1 \implies -45(15B) - 3B = 1 \implies -675B - 3B = 1 \implies -678B = 1 \implies B = -\frac{1}{678}\)
     - \(C = 15B = -\frac{15}{678} = -\frac{5}{226}\)

     Therefore, the particular integral for the \(\sin(3x)\) term is:
     \[
     y_{p2} = -\frac{1}{678}\cos(3x) - \frac{5}{226}\sin(3x)
     \]

### Final Particular Integral:
\[
y_p = y_{p1} + y_{p2} = \frac{1}{18} e^{2x} - \frac{1}{678}\cos(3x) - \frac{5}{226}\sin(3x)
\]

This is the particular integral for the given differential equation.

Let me know if you'd like any further details or have any questions!

### 5 Related Questions:
1. How do we find the complementary function for the given differential equation?
2. Can you explain the method of undetermined coefficients in more detail?
3. How do we apply the annihilator method to this type of problem?
4. What is the difference between homogeneous and non-homogeneous differential equations?
5. How would the solution change if the right-hand side were \(e^{2x} + \cos(3x)\) instead of \(\sin(3x)\)?

### Tip:
For non-homogeneous linear differential equations, always check the form of the non-homogeneous term to choose an appropriate trial solution for the particular integral.

Find the particular integral for the differential equation \( \frac{d^3y}{dx^3} + 2\frac{d^2y}{dx^2} + \frac{dy}{dx} = e^{2x} + \sin(3x) \).

Learn how to find the particular integral of the differential equation \( \frac{d^3y}{dx^3} + 2\frac{d^2y}{dx^2} + \frac{dy}{dx} = e^{2x} + \sin(3x) \) using the method of undetermined coefficients. The solution covers the form of the particular integral for each term on the right-hand side.

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