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To solve the differential equation:

\[
(D^3 - 2D^2 - 5D + 6)y = e^{3x} + 8
\]

We need to find the particular integral (PI). Here are the steps:

---

### Step 1: Analyze the differential operator and input
The differential operator is:

\[
D^3 - 2D^2 - 5D + 6
\]

The right-hand side contains two terms: \( e^{3x} \) and \( 8 \).

---

### Step 2: Solve for \( e^{3x} \) contribution
For \( e^{3x} \), substitute \( D = 3 \) into the operator:

\[
(3)^3 - 2(3)^2 - 5(3) + 6 = 27 - 18 - 15 + 6 = 0
\]

Since the operator annihilates \( e^{3x} \), the method of undetermined coefficients suggests a solution of the form:

\[
y_1 = x \cdot C \cdot e^{3x}
\]

Substitute \( y_1 = xAe^{3x} \) into the equation to determine \( A \).

---

### Step 3: Solve for constant term \( 8 \)
The constant \( 8 \) corresponds to a right-hand side constant. Substitute \( D = 0 \) into the operator:

\[
(0)^3 - 2(0)^2 - 5(0) + 6 = 6
\]

The particular integral for the constant term \( 8 \) is:

\[
y_2 = \frac{8}{6} = \frac{4}{3}
\]

---

### Step 4: Combine results
The total particular integral is:

\[
y = x \cdot \frac{1}{10} e^{3x} + \frac{4}{3}
\]

---

### Correct Answer:
The correct option is:

\[
\boxed{\frac{x}{10} e^{3x} + \frac{4}{3}}
\]

---

Do you want further clarification on this process? Let me know!

### Expanding Questions:
1. How do you verify a particular integral in a differential equation?
2. Why is \( xAe^{3x} \) used when \( e^{3x} \) is annihilated by the operator?
3. What happens if the operator does not annihilate a term in the RHS?
4. How does substituting \( D \) simplify the computation process?
5. Can this method be extended to more complex differential equations?

### Tip:
Always check if the right-hand side term is a root of the auxiliary equation before proposing the form of the particular integral.

Choose the correct particular Integral for the Differential equation \((D^3 - 2D^2 - 5D + 6)y = e^{3x} + 8\)

This problem involves solving a differential equation using the method of undetermined coefficients to find the particular integral for (D^3 - 2D^2 - 5D + 6)y = e^{3x} + 8. Includes step-by-step calculation of terms involving e^{3x} and constants.

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