3. Find the correct form of x(s) and stop. State x(s) as a single fraction. x" +7x' + 6x = 2t x(0) = 1, x'(0) = 2
3. Find the correct form of x(s) and stop. State x(s) as a single fraction. x" +7x' + 6x = 2t x(0) = 1, x'(0) = 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 3: Differential Equation Solution
**Task:** Find the correct form of \( x(s) \) and stop. State \( x(s) \) as a single fraction.
#### Given Differential Equation:
\[ x'' + 7x' + 6x = 2t \]
#### Initial Conditions:
\[ x(0) = -1, \quad x'(0) = 2 \]
### Instructions:
1. **Identify the differential equation along with initial conditions.**
2. **Convert the given differential equation to the Laplace domain if necessary.**
3. **Solve for \( x(s) \).**
4. **Express \( x(s) \) as a single fraction.**
### Steps for solving:
1. **Transform differential equation using Laplace Transforms where appropriate.**
2. **Apply initial conditions.**
3. **Simplify and solve for \( x(s) \).**
### Notes:
- Ensure all steps follow the principles of differential equations and Laplace transformations.
- The result should be simplified to a single fraction to ensure clarity and precision in the solution.
### Visualization:
There are no additional graphs or diagrams provided with this problem, so a detailed step-by-step algebraic process is required to reach the solution.
**End of transcription.**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5dfcde0a-cc8e-4f7e-b5e2-9ec6ad36a3c9%2F540923e4-f9a9-49e5-9d09-d6ed5ede63c6%2Fhjo1tpg_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 3: Differential Equation Solution
**Task:** Find the correct form of \( x(s) \) and stop. State \( x(s) \) as a single fraction.
#### Given Differential Equation:
\[ x'' + 7x' + 6x = 2t \]
#### Initial Conditions:
\[ x(0) = -1, \quad x'(0) = 2 \]
### Instructions:
1. **Identify the differential equation along with initial conditions.**
2. **Convert the given differential equation to the Laplace domain if necessary.**
3. **Solve for \( x(s) \).**
4. **Express \( x(s) \) as a single fraction.**
### Steps for solving:
1. **Transform differential equation using Laplace Transforms where appropriate.**
2. **Apply initial conditions.**
3. **Simplify and solve for \( x(s) \).**
### Notes:
- Ensure all steps follow the principles of differential equations and Laplace transformations.
- The result should be simplified to a single fraction to ensure clarity and precision in the solution.
### Visualization:
There are no additional graphs or diagrams provided with this problem, so a detailed step-by-step algebraic process is required to reach the solution.
**End of transcription.**
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