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–3x Show that the function y= e²* + e* satisfy the differential equation y"+y'-6y=0

–3x Show that the function y= e²* + e* satisfy the differential equation y"+y'-6y=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 Statement** Show that the function \( y = e^{2x} + e^{-3x} \) satisfies the differential equation \( y'' + y' - 6y = 0 \). **Explanation** This problem requires verifying that the given function \( y = e^{2x} + e^{-3x} \) satisfies the given second-order linear differential equation. The equation involves the second derivative \( y'' \), the first derivative \( y' \), and the function \( y \) itself. **Steps to Solve** 1. **Differentiate the function**: - First derivative: \( y' = \frac{d}{dx}(e^{2x} + e^{-3x}) \). - Second derivative: \( y'' = \frac{d^2}{dx^2}(e^{2x} + e^{-3x}) \). 2. **Substitute into the differential equation**: - Substitute \( y \), \( y' \), and \( y'' \) into \( y'' + y' - 6y = 0 \). 3. **Verify equality**: - Simplify the expression to verify the left-hand side equals zero. **Conclusion** The solution concludes by confirming that the function satisfies the differential equation through these calculated derivatives and substitution. This process exemplifies verifying solutions to differential equations, critical in various fields such as physics, engineering, and applied mathematics.
Transcribed Image Text:**Problem Statement** Show that the function \( y = e^{2x} + e^{-3x} \) satisfies the differential equation \( y'' + y' - 6y = 0 \). **Explanation** This problem requires verifying that the given function \( y = e^{2x} + e^{-3x} \) satisfies the given second-order linear differential equation. The equation involves the second derivative \( y'' \), the first derivative \( y' \), and the function \( y \) itself. **Steps to Solve** 1. **Differentiate the function**: - First derivative: \( y' = \frac{d}{dx}(e^{2x} + e^{-3x}) \). - Second derivative: \( y'' = \frac{d^2}{dx^2}(e^{2x} + e^{-3x}) \). 2. **Substitute into the differential equation**: - Substitute \( y \), \( y' \), and \( y'' \) into \( y'' + y' - 6y = 0 \). 3. **Verify equality**: - Simplify the expression to verify the left-hand side equals zero. **Conclusion** The solution concludes by confirming that the function satisfies the differential equation through these calculated derivatives and substitution. This process exemplifies verifying solutions to differential equations, critical in various fields such as physics, engineering, and applied mathematics.
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