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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Question
What was the original differential equation if the general solution for a homogeneous differential equation with constant coefficients is given? Please show all steps.
![Here is the transcription of the mathematical expression in the image:
\[ y(x) = C_1 + C_2 e^{2x} + C_3 x e^{2x} - e^{-x} [C_4 \cos(3x) + C_5 \sin(3x)] \]
Explanation:
This expression defines a function \( y(x) \) which is a linear combination of exponential and trigonometric functions. It includes:
1. A constant term \( C_1 \).
2. An exponential term \( C_2 e^{2x} \).
3. A product of \( x \) and an exponential, \( C_3 x e^{2x} \).
4. A damped trigonometric expression \( - e^{-x} [C_4 \cos(3x) + C_5 \sin(3x)] \), where:
- \( e^{-x} \) acts as a damping factor.
- \( C_4 \cos(3x) \) and \( C_5 \sin(3x) \) are sinusoidal components.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F40174750-f49c-418a-99e5-b99a47847661%2Fd1fdc702-1f7a-4820-8a9f-d3d696bcb404%2Fadtpw1d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Here is the transcription of the mathematical expression in the image:
\[ y(x) = C_1 + C_2 e^{2x} + C_3 x e^{2x} - e^{-x} [C_4 \cos(3x) + C_5 \sin(3x)] \]
Explanation:
This expression defines a function \( y(x) \) which is a linear combination of exponential and trigonometric functions. It includes:
1. A constant term \( C_1 \).
2. An exponential term \( C_2 e^{2x} \).
3. A product of \( x \) and an exponential, \( C_3 x e^{2x} \).
4. A damped trigonometric expression \( - e^{-x} [C_4 \cos(3x) + C_5 \sin(3x)] \), where:
- \( e^{-x} \) acts as a damping factor.
- \( C_4 \cos(3x) \) and \( C_5 \sin(3x) \) are sinusoidal components.
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