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8. Show that the function y = ex (sin 2x + cos 2x) is a solution of the equation y" — 2y' + 5y = 0.

8. Show that the function y = ex (sin 2x + cos 2x) is a solution of the equation y" — 2y' + 5y = 0.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Problem 8: Differential Equation Exploration** Objective: Demonstrate that the function \( y = e^x (\sin 2x + \cos 2x) \) satisfies the differential equation \( y'' - 2y' + 5y = 0 \). ### Approach: 1. **Identify the Function:** - Given: \( y = e^x (\sin 2x + \cos 2x) \) 2. **Compute Derivatives:** - First derivative, \( y' \) - Second derivative, \( y'' \) 3. **Substitute into the Differential Equation:** - Plug \( y \), \( y' \), and \( y'' \) into \( y'' - 2y' + 5y = 0 \) 4. **Verify Solution:** - Simplify the expression to confirm the equation equals zero. ### Explanation: This exercise involves calculus techniques: differentiation and verification by substitution. By following these steps, you validate that the given function is indeed a solution to the specified differential equation.
Transcribed Image Text:**Problem 8: Differential Equation Exploration** Objective: Demonstrate that the function \( y = e^x (\sin 2x + \cos 2x) \) satisfies the differential equation \( y'' - 2y' + 5y = 0 \). ### Approach: 1. **Identify the Function:** - Given: \( y = e^x (\sin 2x + \cos 2x) \) 2. **Compute Derivatives:** - First derivative, \( y' \) - Second derivative, \( y'' \) 3. **Substitute into the Differential Equation:** - Plug \( y \), \( y' \), and \( y'' \) into \( y'' - 2y' + 5y = 0 \) 4. **Verify Solution:** - Simplify the expression to confirm the equation equals zero. ### Explanation: This exercise involves calculus techniques: differentiation and verification by substitution. By following these steps, you validate that the given function is indeed a solution to the specified differential equation.
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