Use the method of undetermined coefficients to determine the form of a particular solution for the given equation.
y triple prime plus 8 y double prime minus 9 yequalsx e Superscript x Baseline plus 11
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Part 1
What is the form of the particular solution with undetermined​ coefficients?
y Subscript p Baseline left parenthesis x right parenthesis equals

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​(Do not use​ d, D,​ e, E,​ i, or I as arbitrary constants since these letters already have defined​ meanings.)

To determine the form of a particular solution \( y_p(x) \) using the method of undetermined coefficients, let’s analyze the non-homogeneous term on the right side of the differential equation:
\[
x e^x + 11.
\]

### Step 1: Determine the form based on each term separately
1. **Term \( x e^x \)**: This term suggests we try a particular solution involving terms of the form \( (Ax + B)e^x \), since \( e^x \) is involved. The factor \( x \) indicates that we will need to account for polynomial terms multiplied by \( e^x \).

2. **Term \( 11 \)**: This is a constant term, so it suggests we try a constant term in the solution, which we can represent as \( C \).

### Step 2: Identify the complete form for \( y_p(x) \)
Because \( x e^x \) includes both \( x e^x \) and \( e^x \) terms, and we also have the constant \( 11 \), the particular solution will include both polynomial terms with \( e^x \) and a constant term.

The complete form of the particular solution for \( y_p(x) \) is:
\[
y_p(x) = (Ax + B)e^x + C.
\]

Here:
- \( (Ax + B)e^x \) accounts for the \( x e^x \) term.
- \( C \) accounts for the constant term \( 11 \).

### Answer
\[
y_p(x) = (Ax + B)e^x + C.
\]

Would you like more details on the undetermined coefficients method?

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### Additional Practice Questions
1. How would the form change if the right side were \( x^2 e^x + 11 \)?
2. What if the differential equation instead had \( e^x \) on the right side without \( x \)?
3. How would you determine the particular solution if the equation were \( y''' + 8y'' - 9y = x^2 e^x \)?
4. If the homogeneous solution already contains \( e^x \), how does that affect \( y_p(x) \)?
5. What is the general form for a non-homogeneous term involving \( e^{2x} \)?

### Tip:
When using undetermined coefficients, always examine the terms on the right side individually to identify the form of each term's particular solution.

Use the method of undetermined coefficients to determine the form of a particular solution for the given equation. y''' + 8y'' - 9y = x e^x + 11. What is the form of the particular solution with undetermined coefficients?

Learn how to find the form of a particular solution for the differential equation y''' + 8y'' - 9y = x e^x + 11 using the method of undetermined coefficients. This solution covers core concepts in differential equations and provides a step-by-step analysis.

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