Calculate the Wronskian. W[ex, ex, cos x, sin x] = -8 (Simplify your answer.) Therefore, the set of functions {ex, ex, cos x, sin x) forms a fundamental solution set for the differential equation y(4) - y = 0. How can a general solution be determined from the fundamental solution set? O A. Express the fundamental solution set as the sum of each particular solution times the value of the Wronskian. O B. Express the fundamental solution set as the sum of each particular solution. O C. Express the fundamental solution set as the product of each particular solution. O D. Express the fundamental solution set as a linear combination using arbitrary constants.
Calculate the Wronskian. W[ex, ex, cos x, sin x] = -8 (Simplify your answer.) Therefore, the set of functions {ex, ex, cos x, sin x) forms a fundamental solution set for the differential equation y(4) - y = 0. How can a general solution be determined from the fundamental solution set? O A. Express the fundamental solution set as the sum of each particular solution times the value of the Wronskian. O B. Express the fundamental solution set as the sum of each particular solution. O C. Express the fundamental solution set as the product of each particular solution. O D. Express the fundamental solution set as a linear combination using arbitrary constants.
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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![### Calculating the Wronskian
Given the set of functions \(\{e^x, e^{-x}, \cos x, \sin x\}\), we are asked to calculate the Wronskian.
\[ W[e^x, e^{-x}, \cos x, \sin x] = -8 \]
This equation has been simplified to give the value \(-8\).
Therefore, the set of functions \(\{e^x, e^{-x}, \cos x, \sin x\}\) forms a fundamental solution set for the differential equation \(y^{(4)} - y = 0\).
**How can a general solution be determined from the fundamental solution set?**
1. **Option A:** Express the fundamental solution set as the sum of each particular solution times the value of the Wronskian.
2. **Option B:** Express the fundamental solution set as the sum of each particular solution.
3. **Option C:** Express the fundamental solution set as the product of each particular solution.
4. **Option D:** Express the fundamental solution set as a linear combination using arbitrary constants.
The correct method to determine a general solution is typically to use option D: expressing the fundamental solution set as a linear combination using arbitrary constants. This allows for the formation of a more general solution to the differential equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbbba0909-7b4f-4e80-8e24-06dfda2d061e%2Fd47ae1ac-3b77-45f3-bf52-098ef7f36908%2Fbl8lq9r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculating the Wronskian
Given the set of functions \(\{e^x, e^{-x}, \cos x, \sin x\}\), we are asked to calculate the Wronskian.
\[ W[e^x, e^{-x}, \cos x, \sin x] = -8 \]
This equation has been simplified to give the value \(-8\).
Therefore, the set of functions \(\{e^x, e^{-x}, \cos x, \sin x\}\) forms a fundamental solution set for the differential equation \(y^{(4)} - y = 0\).
**How can a general solution be determined from the fundamental solution set?**
1. **Option A:** Express the fundamental solution set as the sum of each particular solution times the value of the Wronskian.
2. **Option B:** Express the fundamental solution set as the sum of each particular solution.
3. **Option C:** Express the fundamental solution set as the product of each particular solution.
4. **Option D:** Express the fundamental solution set as a linear combination using arbitrary constants.
The correct method to determine a general solution is typically to use option D: expressing the fundamental solution set as a linear combination using arbitrary constants. This allows for the formation of a more general solution to the differential equation.
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