Two linearly independent solutions of the differential equation y" – 4y' + 4y = 0 are O yı = e2x, y2 = e2x O yı = e2x, y2 = xe2x O yi = e2x, y2 = e-2x O yi = e-2x, y2 = xe2x
Two linearly independent solutions of the differential equation y" – 4y' + 4y = 0 are O yı = e2x, y2 = e2x O yı = e2x, y2 = xe2x O yi = e2x, y2 = e-2x O yi = e-2x, y2 = xe2x
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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![**Question:**
Two linearly independent solutions of the differential equation \( y'' - 4y' + 4y = 0 \) are
- \( y_1 = e^{2x}, \quad y_2 = e^{2x} \)
- \( y_1 = e^{2x}, \quad y_2 = xe^{2x} \)
- \( y_1 = e^{2x}, \quad y_2 = e^{-2x} \)
- \( y_1 = e^{-2x}, \quad y_2 = xe^{2x} \)
**Explanation:**
This question involves finding linearly independent solutions to a second-order linear homogeneous differential equation. The differential equation given is:
\[ y'' - 4y' + 4y = 0 \]
The correct pair of linearly independent solutions should satisfy the equation and provide a fundamental set of solutions, which can be combined to form the general solution.
The potential solutions provided include exponential functions and combinations involving products of \( e^{2x} \) and powers of \( x \). The correct solution would potentially involve recognizing the characteristic equation related to the differential equation and finding roots that provide distinct solutions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F74a43f44-8ce4-41e9-89f6-5a77f02dff59%2F3bb13d41-2e43-4874-9576-2f49ceac18c3%2Fwt8kodn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question:**
Two linearly independent solutions of the differential equation \( y'' - 4y' + 4y = 0 \) are
- \( y_1 = e^{2x}, \quad y_2 = e^{2x} \)
- \( y_1 = e^{2x}, \quad y_2 = xe^{2x} \)
- \( y_1 = e^{2x}, \quad y_2 = e^{-2x} \)
- \( y_1 = e^{-2x}, \quad y_2 = xe^{2x} \)
**Explanation:**
This question involves finding linearly independent solutions to a second-order linear homogeneous differential equation. The differential equation given is:
\[ y'' - 4y' + 4y = 0 \]
The correct pair of linearly independent solutions should satisfy the equation and provide a fundamental set of solutions, which can be combined to form the general solution.
The potential solutions provided include exponential functions and combinations involving products of \( e^{2x} \) and powers of \( x \). The correct solution would potentially involve recognizing the characteristic equation related to the differential equation and finding roots that provide distinct solutions.
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