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:**
Determine if the function \( y(x) = 3\sin(2x) + e^{-x} \) is a solution to the second-order differential equation:
\[ y'' + 4y = 5e^{-x} \]
**Explanation:**
To verify that the given function is a solution to the differential equation, follow these steps:
1. **Calculate the first derivative \( y'(x) \) of the function \( y(x) = 3\sin(2x) + e^{-x} \).**
2. **Calculate the second derivative \( y''(x) \).**
3. **Substitute \( y(x) \) and \( y''(x) \) into the differential equation \( y'' + 4y = 5e^{-x} \).**
4. **Simplify the expression to check if both sides of the equation are equal.**
If both sides of the equation are equal after simplification, then \( y(x) = 3\sin(2x) + e^{-x} \) is a solution to the differential equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F24fae4ed-b1b1-4171-bba8-4d29349ae5e5%2F4fac4e42-9df3-4cb7-8e33-858420e1f91f%2F9bbg1u7_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Determine if the function \( y(x) = 3\sin(2x) + e^{-x} \) is a solution to the second-order differential equation:
\[ y'' + 4y = 5e^{-x} \]
**Explanation:**
To verify that the given function is a solution to the differential equation, follow these steps:
1. **Calculate the first derivative \( y'(x) \) of the function \( y(x) = 3\sin(2x) + e^{-x} \).**
2. **Calculate the second derivative \( y''(x) \).**
3. **Substitute \( y(x) \) and \( y''(x) \) into the differential equation \( y'' + 4y = 5e^{-x} \).**
4. **Simplify the expression to check if both sides of the equation are equal.**
If both sides of the equation are equal after simplification, then \( y(x) = 3\sin(2x) + e^{-x} \) is a solution to the differential equation.
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