Use the variation of parameters formula to find a general solution of the system x'(t) = Ax(t) + f(t), where A and f(t) are given. 19 A = f(t) = 17 9 x(t) = O
Use the variation of parameters formula to find a general solution of the system x'(t) = Ax(t) + f(t), where A and f(t) are given. 19 A = f(t) = 17 9 x(t) = O
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem Statement:**
Use the variation of parameters formula to find a general solution of the system \( x'(t) = Ax(t) + f(t) \), where \( A \) and \( f(t) \) are given.
**Matrix and Vector Definitions:**
\[
A = \begin{bmatrix} 1 & 9 \\ 17 & 9 \end{bmatrix}, \quad f(t) = \begin{bmatrix} 5 \\ -5 \end{bmatrix}
\]
**Objective:**
Determine \( x(t) = \begin{bmatrix} \_ \\ \_ \end{bmatrix} \).
**Explanation:**
The system \( x'(t) = Ax(t) + f(t) \) is a linear non-homogeneous differential equation. The matrix \( A \) is a \( 2 \times 2 \) matrix, and \( f(t) \) is a \( 2 \times 1 \) vector. The solution involves finding a particular solution using the method of variation of parameters along with the complementary (homogeneous) solution. To solve, evaluate and integrate using these elements to achieve the general solution vector \( x(t) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b9e45d9-979e-4aa9-8971-5ebb0f6642d4%2Fb12b03c7-88a5-4942-8b1b-8c39c9254ead%2Fg801pke_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Use the variation of parameters formula to find a general solution of the system \( x'(t) = Ax(t) + f(t) \), where \( A \) and \( f(t) \) are given.
**Matrix and Vector Definitions:**
\[
A = \begin{bmatrix} 1 & 9 \\ 17 & 9 \end{bmatrix}, \quad f(t) = \begin{bmatrix} 5 \\ -5 \end{bmatrix}
\]
**Objective:**
Determine \( x(t) = \begin{bmatrix} \_ \\ \_ \end{bmatrix} \).
**Explanation:**
The system \( x'(t) = Ax(t) + f(t) \) is a linear non-homogeneous differential equation. The matrix \( A \) is a \( 2 \times 2 \) matrix, and \( f(t) \) is a \( 2 \times 1 \) vector. The solution involves finding a particular solution using the method of variation of parameters along with the complementary (homogeneous) solution. To solve, evaluate and integrate using these elements to achieve the general solution vector \( x(t) \).
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