pruttes) Select either a) or b) to solve. Please indicate which problem you sol Solve the IVP using the method of Laplace transforms y"-2y'+y=2t-3; y(1) = 5, y'(1)=3 =3, y'(0)=0..
pruttes) Select either a) or b) to solve. Please indicate which problem you sol Solve the IVP using the method of Laplace transforms y"-2y'+y=2t-3; y(1) = 5, y'(1)=3 =3, y'(0)=0..
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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![### Educational Transcription and Explanation of Laplace Transform Problem Solving
#### Problem Statement:
- **Select either a) or b) to solve. Please indicate which problem you solved.**
- a) Solve the IVP using the method of Laplace transforms: \( y^{\prime \prime} - 2y^{\prime} + y = 2 - 3 \)
- \( y(1) = 5 \), \( y^{\prime}(1) = 3 \)
- b) Solve the IVP using the method of Laplace transforms: \( y^{\prime \prime} - 2y^{\prime} + 8y = 0 \)
- \( y(0) = 3 \), \( y^{\prime}(0) = 0 \)
#### Solution:
1. **Given Formula:**
\[ n = t + 1 \]
\[ t = n + 1 \]
\[ y''(n + 1) - 2y'(n + 1) + y(n + 1) = 2(n + 1) - 3 \]
2. **Transform and Initial Conditions:**
\[ u(t) = y(n+1) \]
\[ u'(t) = y'(n + 1) \]
\[ y(0) = 5 \quad \text{(Given)} \]
\[ y'(0) = 3 \quad \text{(Given)} \]
\[ u(0) = y(1) = 5 \quad (y_0 = 5) \]
\[ u_1(0) = y(1) = 3 \quad (u_0 = 3) \]
3. **Differential Transform:**
\[ u'' - 2u' + u = 2n - 1 \]
4. **Initial Values for Transformation:**
\[ u(0) = 5 \]
\[ u'(0) = 3 \]
5. **Laplace Transformation:**
\[ (s^2 - 2s + 1) u = 2u(s) / s^2 + 3 \]
6. **Solving the Transformed Equation:**
\[ \hat{U}(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F946e5b6f-098d-47d1-a6b1-15872ebd1054%2Fbe1039dd-c390-4586-aa12-dec4b2392ebf%2Fp60pww_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Educational Transcription and Explanation of Laplace Transform Problem Solving
#### Problem Statement:
- **Select either a) or b) to solve. Please indicate which problem you solved.**
- a) Solve the IVP using the method of Laplace transforms: \( y^{\prime \prime} - 2y^{\prime} + y = 2 - 3 \)
- \( y(1) = 5 \), \( y^{\prime}(1) = 3 \)
- b) Solve the IVP using the method of Laplace transforms: \( y^{\prime \prime} - 2y^{\prime} + 8y = 0 \)
- \( y(0) = 3 \), \( y^{\prime}(0) = 0 \)
#### Solution:
1. **Given Formula:**
\[ n = t + 1 \]
\[ t = n + 1 \]
\[ y''(n + 1) - 2y'(n + 1) + y(n + 1) = 2(n + 1) - 3 \]
2. **Transform and Initial Conditions:**
\[ u(t) = y(n+1) \]
\[ u'(t) = y'(n + 1) \]
\[ y(0) = 5 \quad \text{(Given)} \]
\[ y'(0) = 3 \quad \text{(Given)} \]
\[ u(0) = y(1) = 5 \quad (y_0 = 5) \]
\[ u_1(0) = y(1) = 3 \quad (u_0 = 3) \]
3. **Differential Transform:**
\[ u'' - 2u' + u = 2n - 1 \]
4. **Initial Values for Transformation:**
\[ u(0) = 5 \]
\[ u'(0) = 3 \]
5. **Laplace Transformation:**
\[ (s^2 - 2s + 1) u = 2u(s) / s^2 + 3 \]
6. **Solving the Transformed Equation:**
\[ \hat{U}(
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