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
Solve the initial-value problem given by the differential equation:
\[ x' = Ax, \quad x(0) = x_0 \]
where the matrix \( A \) and the initial condition \( x(0) \) are specified as follows:
\[ A = \begin{bmatrix} 2 & -5 \\ 4 & -2 \end{bmatrix}, \quad x(0) = \begin{bmatrix} 2 \\ 3 \end{bmatrix} \]
### Explanation
This problem involves solving a system of linear differential equations with a given initial value. The solution approach typically involves finding the eigenvalues and eigenvectors of matrix \( A \), constructing the general solution in terms of these eigenvectors, and applying the initial condition to find the specific solution.
This is a fundamental concept in linear algebra and differential equations, often encountered in engineering and physical sciences. The matrix \( A \) represents a linear transformation that affects how the system evolves over time, and \( x(0) \) represents the state of the system at time \( t = 0 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F86847a45-621f-4eb2-b143-0cde4fac410f%2F02013788-b7cb-45be-af3c-ac8c2d641bae%2F1jtrmw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Solve the initial-value problem given by the differential equation:
\[ x' = Ax, \quad x(0) = x_0 \]
where the matrix \( A \) and the initial condition \( x(0) \) are specified as follows:
\[ A = \begin{bmatrix} 2 & -5 \\ 4 & -2 \end{bmatrix}, \quad x(0) = \begin{bmatrix} 2 \\ 3 \end{bmatrix} \]
### Explanation
This problem involves solving a system of linear differential equations with a given initial value. The solution approach typically involves finding the eigenvalues and eigenvectors of matrix \( A \), constructing the general solution in terms of these eigenvectors, and applying the initial condition to find the specific solution.
This is a fundamental concept in linear algebra and differential equations, often encountered in engineering and physical sciences. The matrix \( A \) represents a linear transformation that affects how the system evolves over time, and \( x(0) \) represents the state of the system at time \( t = 0 \).
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