A particular 1 3 a. b. C. d. e. solution of y" + 2y'-3y = 1 + xex. 1 3 1 1 2 --xx + 1x2²ex 3 8 8 3 -1/33 1 | M | M xex+ 2 --xx-x²ex 8 1 16 1 16 −13+ ·x² ex 3 -xex + 1x² e 2 ex -xex + 1x² ex
A particular 1 3 a. b. C. d. e. solution of y" + 2y'-3y = 1 + xex. 1 3 1 1 2 --xx + 1x2²ex 3 8 8 3 -1/33 1 | M | M xex+ 2 --xx-x²ex 8 1 16 1 16 −13+ ·x² ex 3 -xex + 1x² e 2 ex -xex + 1x² ex
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![### A particular solution of \( y'' + 2y' - 3y = 1 + xe^x \).
#### Options:
**a.**
\[
\frac{1}{3} - \frac{1}{4} xe^x + \frac{3}{4} x^2 e^x
\]
**b.**
\[
-\frac{1}{3} - \frac{2}{8} xe^x + \frac{1}{8} x^2 e^x
\]
**c.**
\[
-\frac{1}{3} - \frac{2}{8} xe^x - \frac{3}{8} x^2 e^x
\]
**d.**
\[
-\frac{1}{3} - \frac{1}{16} xe^x + \frac{1}{8} x^2 e^x
\]
**e.**
\[
-\frac{1}{3} + \frac{1}{16} xe^x + \frac{1}{8} x^2 e^x
\]
### Explanation:
In this problem, you are asked to identify the particular solution to the non-homogeneous linear differential equation \( y'' + 2y' - 3y = 1 + xe^x \).
#### Breakdown of Options:
- Each option represents a different candidate for the particular solution of the differential equation.
- The terms in each option involve combinations of constants, exponential functions \(e^x\), and their products with \(x\) and \(x^2\).
For educational synthesis:
- **Constants**: The constants could represent specific values that arise from the solution process.
- **Exponential Terms**: The terms involving \( xe^x \) and \( x^2e^x \) suggest the use of the method of undetermined coefficients to account for the non-homogeneous term on the right-hand side of the equation \( 1 + xe^x \).
Understanding which particular solution correctly satisfies the given differential equation is often achieved through substitution and verification.
**Step-by-step approach**:
1. **Assumption**: Assume a particular form based on the non-homogeneous term.
2. **Substitute**: Substitute this assumed form into the differential equation.
3. **Solve for coefficients**: Balance the equation by solving for unknown](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc59b3e01-f4fe-4ed0-a002-b05c91db4102%2F6d967d61-b872-4fcc-8fcf-c8bcaa1f7621%2Fpnba8fd_processed.png&w=3840&q=75)
Transcribed Image Text:### A particular solution of \( y'' + 2y' - 3y = 1 + xe^x \).
#### Options:
**a.**
\[
\frac{1}{3} - \frac{1}{4} xe^x + \frac{3}{4} x^2 e^x
\]
**b.**
\[
-\frac{1}{3} - \frac{2}{8} xe^x + \frac{1}{8} x^2 e^x
\]
**c.**
\[
-\frac{1}{3} - \frac{2}{8} xe^x - \frac{3}{8} x^2 e^x
\]
**d.**
\[
-\frac{1}{3} - \frac{1}{16} xe^x + \frac{1}{8} x^2 e^x
\]
**e.**
\[
-\frac{1}{3} + \frac{1}{16} xe^x + \frac{1}{8} x^2 e^x
\]
### Explanation:
In this problem, you are asked to identify the particular solution to the non-homogeneous linear differential equation \( y'' + 2y' - 3y = 1 + xe^x \).
#### Breakdown of Options:
- Each option represents a different candidate for the particular solution of the differential equation.
- The terms in each option involve combinations of constants, exponential functions \(e^x\), and their products with \(x\) and \(x^2\).
For educational synthesis:
- **Constants**: The constants could represent specific values that arise from the solution process.
- **Exponential Terms**: The terms involving \( xe^x \) and \( x^2e^x \) suggest the use of the method of undetermined coefficients to account for the non-homogeneous term on the right-hand side of the equation \( 1 + xe^x \).
Understanding which particular solution correctly satisfies the given differential equation is often achieved through substitution and verification.
**Step-by-step approach**:
1. **Assumption**: Assume a particular form based on the non-homogeneous term.
2. **Substitute**: Substitute this assumed form into the differential equation.
3. **Solve for coefficients**: Balance the equation by solving for unknown
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