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; {e*, e¯X, cos x, sin x} 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 We*, e -*, cos x, sin x] # В. Verify that each function satisfies the given differential equation and then verify that We*, e -*, coS x, sin x =
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; {e*, e¯X, cos x, sin x} 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 We*, e -*, cos x, sin x] # В. Verify that each function satisfies the given differential equation and then verify that We*, e -*, coS x, sin x =
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb4c9d77e-368b-42fe-b276-8a66252e95fb%2F0d39f146-11f6-4d1b-9151-5b614453ab19%2F8dk1sq_processed.png&w=3840&q=75)
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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