The eigenvalues in this problem are all nonnegative. First determine whether λ = 0 is an eigenvalue; then find the positive eigenvalues and associated eigenfunctions. y' +42y=0; y'(0) = 0, y(1)=0 Is λ = 0 an eigenvalue of this problem? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. No, when A=0 the only solution to the given equation is the trivial solution. OB. Yes, λ = 0 is an eigenvalue with corresponding eigenfunction yo(x) = OC. No, when A=0 there are an infinite number of nontrivial solutions to the given equation. What are the positive eigenvalues and associated eigenfunctions yn(x) for n = 1, 2, 3, ...? Select the correct choice below and fill in the answer boxes to complete your choice. (Type exact answers.) OA. The positive eigenvalues are λ = OB. The positive eigenvalues are λ = The eigenfunctions are yn(x) = The eigenfunctions are yn(x) = regardless of whether n is even or odd. when n is even and yn(x) = when n is odd.
The eigenvalues in this problem are all nonnegative. First determine whether λ = 0 is an eigenvalue; then find the positive eigenvalues and associated eigenfunctions. y' +42y=0; y'(0) = 0, y(1)=0 Is λ = 0 an eigenvalue of this problem? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. No, when A=0 the only solution to the given equation is the trivial solution. OB. Yes, λ = 0 is an eigenvalue with corresponding eigenfunction yo(x) = OC. No, when A=0 there are an infinite number of nontrivial solutions to the given equation. What are the positive eigenvalues and associated eigenfunctions yn(x) for n = 1, 2, 3, ...? Select the correct choice below and fill in the answer boxes to complete your choice. (Type exact answers.) OA. The positive eigenvalues are λ = OB. The positive eigenvalues are λ = The eigenfunctions are yn(x) = The eigenfunctions are yn(x) = regardless of whether n is even or odd. when n is even and yn(x) = when n is odd.
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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![### Eigenvalue Problem Analysis
The eigenvalues in this problem are all nonnegative. First determine whether \(\lambda = 0\) is an eigenvalue; then find the positive eigenvalues and associated eigenfunctions.
#### Given Differential Equation:
\[ y'' + 4\lambda y = 0; \quad y'(0) = 0; \quad y(1) = 0 \]
---
### Step 1: Determine if \(\lambda = 0\) is an Eigenvalue
**Question: Is \(\lambda = 0\) an eigenvalue of this problem? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.**
- **A.** No, when \(\lambda = 0\) the only solution to the given equation is the trivial solution.
- **B.** Yes, \(\lambda = 0\) is an eigenvalue with corresponding eigenfunction \(y_0(x) = \_\_\_\).
- **C.** No, when \(\lambda = 0\) there are an infinite number of nontrivial solutions to the given equation.
**Answer:**
- **A.** No, when \(\lambda = 0\) the only solution to the given equation is the trivial solution.
---
### Step 2: Determine the Positive Eigenvalues and Eigenfunctions
**Question: What are the positive eigenvalues \(\lambda_n\) and associated eigenfunctions \(y_n(x)\) for \(n = 1, 2, 3, \ldots\)? Select the correct choice below and fill in the answer boxes to complete your choice.**
- **A.** The positive eigenvalues are \(\lambda_n = \_\_\_\). The eigenfunctions are \(y_n(x) = \_\_\_\) regardless of whether \(n\) is even or odd.
- **B.** The positive eigenvalues are \(\lambda_n = \_\_\_\). The eigenfunctions are \(y_n(x) = \_\_\_\) when \(n\) is even and \(y_n(x) = \_\_\_\) when \(n\) is odd.
**Answer:**
- **A.** The positive eigenvalues are \(\lambda_n = \_\_\_\). The eigenfunctions are \(y_n(x) = \_\_\_\) regardless of whether \(n\)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F717f4d87-5a32-4cdf-855f-90bb556622e3%2Ff8e10a43-da6e-4176-b453-5ab2b8aec773%2Ft9v6g2p_processed.png&w=3840&q=75)
Transcribed Image Text:### Eigenvalue Problem Analysis
The eigenvalues in this problem are all nonnegative. First determine whether \(\lambda = 0\) is an eigenvalue; then find the positive eigenvalues and associated eigenfunctions.
#### Given Differential Equation:
\[ y'' + 4\lambda y = 0; \quad y'(0) = 0; \quad y(1) = 0 \]
---
### Step 1: Determine if \(\lambda = 0\) is an Eigenvalue
**Question: Is \(\lambda = 0\) an eigenvalue of this problem? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.**
- **A.** No, when \(\lambda = 0\) the only solution to the given equation is the trivial solution.
- **B.** Yes, \(\lambda = 0\) is an eigenvalue with corresponding eigenfunction \(y_0(x) = \_\_\_\).
- **C.** No, when \(\lambda = 0\) there are an infinite number of nontrivial solutions to the given equation.
**Answer:**
- **A.** No, when \(\lambda = 0\) the only solution to the given equation is the trivial solution.
---
### Step 2: Determine the Positive Eigenvalues and Eigenfunctions
**Question: What are the positive eigenvalues \(\lambda_n\) and associated eigenfunctions \(y_n(x)\) for \(n = 1, 2, 3, \ldots\)? Select the correct choice below and fill in the answer boxes to complete your choice.**
- **A.** The positive eigenvalues are \(\lambda_n = \_\_\_\). The eigenfunctions are \(y_n(x) = \_\_\_\) regardless of whether \(n\) is even or odd.
- **B.** The positive eigenvalues are \(\lambda_n = \_\_\_\). The eigenfunctions are \(y_n(x) = \_\_\_\) when \(n\) is even and \(y_n(x) = \_\_\_\) when \(n\) is odd.
**Answer:**
- **A.** The positive eigenvalues are \(\lambda_n = \_\_\_\). The eigenfunctions are \(y_n(x) = \_\_\_\) regardless of whether \(n\)
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