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 4: System of Differential Equations
Solve the system of differential equations consisting of the following equations:
\[ (D - 2)x - y = 0 \]
\[ (D + 5)y + 6x = 0 \]
Where \(D\) is the differential operator with respect to \(t\).
### Explanation
In the given system of differential equations, \( (D - 2)x - y = 0 \) and \( (D + 5)y + 6x = 0 \), we need to find the functions \( x(t) \) and \( y(t) \) that satisfy both equations simultaneously.
Here, the differential operator \( D \) represents differentiation with respect to the variable \( t \). Specifically, for any function \( f(t) \):
\[ Df = \frac{df}{dt} \]
By solving this system, we aim to determine the explicit forms of \( x(t) \) and \( y(t) \).
This problem belongs to the category of linear differential equations with constant coefficients, a common type of problem in advanced mathematics, particularly in the study of dynamic systems and control theory.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd9a90a9d-74be-4386-83ba-9747aab69b9e%2F4ec5aff5-cca3-4a7c-bf47-db7eb3b12e0a%2Fc9wi22e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 4: System of Differential Equations
Solve the system of differential equations consisting of the following equations:
\[ (D - 2)x - y = 0 \]
\[ (D + 5)y + 6x = 0 \]
Where \(D\) is the differential operator with respect to \(t\).
### Explanation
In the given system of differential equations, \( (D - 2)x - y = 0 \) and \( (D + 5)y + 6x = 0 \), we need to find the functions \( x(t) \) and \( y(t) \) that satisfy both equations simultaneously.
Here, the differential operator \( D \) represents differentiation with respect to the variable \( t \). Specifically, for any function \( f(t) \):
\[ Df = \frac{df}{dt} \]
By solving this system, we aim to determine the explicit forms of \( x(t) \) and \( y(t) \).
This problem belongs to the category of linear differential equations with constant coefficients, a common type of problem in advanced mathematics, particularly in the study of dynamic systems and control theory.
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