Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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**Title: Verifying Fundamental Solution Sets Using the Wronskian**
**Problem Statement:**
Using the Wronskian, verify that the given functions form a fundamental solution set for the given differential equation and find a general solution.
\[ y^{(4)} - y = 0; \quad \{e^x, e^{-x}, \cos x, \sin x\} \]
**Question:**
What should be done to verify that the given set of functions forms a fundamental solution set to the given differential equation? Select the correct choice below and fill in the answer box to complete your choice. *(Simplify your answer.)*
- **(A)** Verify that each function satisfies the given differential equation and then verify that \( W[e^x, e^{-x}, \cos x, \sin x] \neq \boxed{\ } \).
- **(B)** Verify that each function satisfies the given differential equation and then verify that \( W[e^x, e^{-x}, \cos x, \sin x] = \boxed{\ } \).
---
**Explanation:**
To verify that the functions form a fundamental solution set for the differential equation \( y^{(4)} - y = 0 \), you need to follow these steps:
1. **Substitution into the Differential Equation:**
Ensure each function individually satisfies the differential equation. Substitute each function into the equation and simplify.
2. **Calculate the Wronskian:**
The Wronskian \( W \) of a set of functions is a determinant that helps in determining linear independence. For the given functions \( \{e^x, e^{-x}, \cos x, \sin x\} \), compute the Wronskian.
3. **Verify the Wronskian Condition:**
- If \( W \neq 0 \) at any point in the interval of consideration, the functions are linearly independent, thus forming a fundamental set of solutions.
- Select the correct option based on whether the Wronskian is zero or not.](https://content.bartleby.com/qna-images/question/b4c9d77e-368b-42fe-b276-8a66252e95fb/0d39f146-11f6-4d1b-9151-5b614453ab19/8dk1sq_processed.png)
Transcribed Image Text:---
**Title: Verifying Fundamental Solution Sets Using the Wronskian**
**Problem Statement:**
Using the Wronskian, verify that the given functions form a fundamental solution set for the given differential equation and find a general solution.
\[ y^{(4)} - y = 0; \quad \{e^x, e^{-x}, \cos x, \sin x\} \]
**Question:**
What should be done to verify that the given set of functions forms a fundamental solution set to the given differential equation? Select the correct choice below and fill in the answer box to complete your choice. *(Simplify your answer.)*
- **(A)** Verify that each function satisfies the given differential equation and then verify that \( W[e^x, e^{-x}, \cos x, \sin x] \neq \boxed{\ } \).
- **(B)** Verify that each function satisfies the given differential equation and then verify that \( W[e^x, e^{-x}, \cos x, \sin x] = \boxed{\ } \).
---
**Explanation:**
To verify that the functions form a fundamental solution set for the differential equation \( y^{(4)} - y = 0 \), you need to follow these steps:
1. **Substitution into the Differential Equation:**
Ensure each function individually satisfies the differential equation. Substitute each function into the equation and simplify.
2. **Calculate the Wronskian:**
The Wronskian \( W \) of a set of functions is a determinant that helps in determining linear independence. For the given functions \( \{e^x, e^{-x}, \cos x, \sin x\} \), compute the Wronskian.
3. **Verify the Wronskian Condition:**
- If \( W \neq 0 \) at any point in the interval of consideration, the functions are linearly independent, thus forming a fundamental set of solutions.
- Select the correct option based on whether the Wronskian is zero or not.
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