B) Use the given general solution to solve the initial value problems. Is there a unique solution, infinitely many solutions, or no solution? 13) (x + 2)y" + (x+1)y' y = 0, a) y(0) = 3, y'(0) = -1. {x+1, e-}. b) y(-2) = 1, y'(-2)=-1. c) y(-2) = 1, y'(-2) = 2.
B) Use the given general solution to solve the initial value problems. Is there a unique solution, infinitely many solutions, or no solution? 13) (x + 2)y" + (x+1)y' y = 0, a) y(0) = 3, y'(0) = -1. {x+1, e-}. b) y(-2) = 1, y'(-2)=-1. c) y(-2) = 1, y'(-2) = 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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![Certainly! Here's the transcription and explanation suitable for an educational website:
---
**Problem B: Initial Value Problems**
Use the given general solution to solve the initial value problems. Determine if there is a unique solution, infinitely many solutions, or no solution.
---
**Equation:**
\[
(x + 2)y'' + (x + 1)y' - y = 0
\]
**General Solution:**
\[
\{x + 1, e^{-x}\}
\]
**Initial Conditions:**
1. **a)** \(y(0) = 3\), \(y'(0) = -1\)
2. **b)** \(y(-2) = 1\), \(y'(-2) = -1\)
3. **c)** \(y(-2) = 1\), \(y'(-2) = 2\)
---
### Explanation
The problem provides an ordinary differential equation with initial conditions and asks for determining the nature of the solutions:
- **Equation Analysis:** The differential equation is second-order linear with variable coefficients.
- **General Solution:** The solution set \(\{x + 1, e^{-x}\}\) suggests two linearly independent solutions to the homogeneous equation.
- **Initial Value Problems:** For each set of initial conditions (a, b, c), substitute into the general solution and its derivatives to check if a unique solution can be determined:
- **Case a:** Solve by substituting \(x = 0\).
- **Case b:** Solve by substituting \(x = -2\).
- **Case c:** Analyze for inconsistencies or determine uniqueness.
**Conclusion:** The problem is about solving the initial value problem and validating the conditions for the existence and uniqueness of solutions based on the given differential equation and initial conditions.
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6d6e0021-e2cc-4c08-9219-eb90d4e8405b%2Faaa936ff-1eb3-4e4c-9686-b2644d362585%2Ftu7l6mah_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Certainly! Here's the transcription and explanation suitable for an educational website:
---
**Problem B: Initial Value Problems**
Use the given general solution to solve the initial value problems. Determine if there is a unique solution, infinitely many solutions, or no solution.
---
**Equation:**
\[
(x + 2)y'' + (x + 1)y' - y = 0
\]
**General Solution:**
\[
\{x + 1, e^{-x}\}
\]
**Initial Conditions:**
1. **a)** \(y(0) = 3\), \(y'(0) = -1\)
2. **b)** \(y(-2) = 1\), \(y'(-2) = -1\)
3. **c)** \(y(-2) = 1\), \(y'(-2) = 2\)
---
### Explanation
The problem provides an ordinary differential equation with initial conditions and asks for determining the nature of the solutions:
- **Equation Analysis:** The differential equation is second-order linear with variable coefficients.
- **General Solution:** The solution set \(\{x + 1, e^{-x}\}\) suggests two linearly independent solutions to the homogeneous equation.
- **Initial Value Problems:** For each set of initial conditions (a, b, c), substitute into the general solution and its derivatives to check if a unique solution can be determined:
- **Case a:** Solve by substituting \(x = 0\).
- **Case b:** Solve by substituting \(x = -2\).
- **Case c:** Analyze for inconsistencies or determine uniqueness.
**Conclusion:** The problem is about solving the initial value problem and validating the conditions for the existence and uniqueness of solutions based on the given differential equation and initial conditions.
---
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