Find a particular solution, y,(x), of y(ª – y" = 12e3 – 5x + 2. -
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem Statement:**
Find a particular solution, \( y_p(x) \), of the differential equation:
\[ y^{(4)} - y'' = 12e^{3x} - 5x + 2. \]
---
**Explanation:**
This problem involves finding a particular solution to the fourth-order linear differential equation. The equation contains an exponential term \( e^{3x} \) and polynomial terms \( x \) and a constant.
The approach to solve this involves using the method of undetermined coefficients or variation of parameters, targeting the non-homogeneous components:
- \( 12e^{3x} \) (exponential term),
- \( -5x + 2 \) (polynomial terms).
**Key Steps for Solution:**
1. **Characteristic Equation:** Identify and solve the characteristic equation of the homogeneous part to find the complementary solution.
2. **Particular Solution \( y_p(x) \):**
- Assume a form for \( y_p(x) \) based on the right-hand side (RHS) of the differential equation.
- Substitute the assumed solution into the differential equation.
- Equate and solve for undetermined coefficients.
3. **General Solution:**
- Combine the particular and complementary solutions to form the general solution.
4. **Verification:**
- Differentiate and substitute back into the original equation to verify correctness.
By following these steps, the particular solution \( y_p(x) \) will be determined, addressing the given differential equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66d256bd-e8fc-4d2e-9d73-ec97867bbef6%2F143f3ffc-5746-47ab-9c1f-8c15736f4e89%2Fwd8bviy6_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find a particular solution, \( y_p(x) \), of the differential equation:
\[ y^{(4)} - y'' = 12e^{3x} - 5x + 2. \]
---
**Explanation:**
This problem involves finding a particular solution to the fourth-order linear differential equation. The equation contains an exponential term \( e^{3x} \) and polynomial terms \( x \) and a constant.
The approach to solve this involves using the method of undetermined coefficients or variation of parameters, targeting the non-homogeneous components:
- \( 12e^{3x} \) (exponential term),
- \( -5x + 2 \) (polynomial terms).
**Key Steps for Solution:**
1. **Characteristic Equation:** Identify and solve the characteristic equation of the homogeneous part to find the complementary solution.
2. **Particular Solution \( y_p(x) \):**
- Assume a form for \( y_p(x) \) based on the right-hand side (RHS) of the differential equation.
- Substitute the assumed solution into the differential equation.
- Equate and solve for undetermined coefficients.
3. **General Solution:**
- Combine the particular and complementary solutions to form the general solution.
4. **Verification:**
- Differentiate and substitute back into the original equation to verify correctness.
By following these steps, the particular solution \( y_p(x) \) will be determined, addressing the given differential equation.
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