Find a particular solution to 3 " sin (3t). 2 -3 5
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:**
Find a particular solution to the following system of differential equations:
\[
\vec{x}'' = \begin{bmatrix} 1 & 3 \\ -3 & 5 \end{bmatrix} \vec{x} + \begin{bmatrix} -1 \\ 2 \end{bmatrix} \sin(3t).
\]
**Explanation:**
This problem involves finding a particular solution to a second-order vector differential equation. The given equation is expressed in matrix form:
- \(\vec{x}''\) represents the second derivative of the vector \(\vec{x}\) with respect to time \(t\).
- The coefficient matrix \(\begin{bmatrix} 1 & 3 \\ -3 & 5 \end{bmatrix}\) operates on the vector \(\vec{x}\).
- The term \(\begin{bmatrix} -1 \\ 2 \end{bmatrix} \sin(3t)\) is the non-homogeneous part, introducing a sinusoidal forcing function.
The goal is to determine the particular vector function \(\vec{x}(t)\) that satisfies this differential equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F31330aa5-a64b-4d2f-80e6-6ea74abb2a34%2F7306f88d-34b8-49e6-95b8-33d3d6316a25%2Fwywile_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find a particular solution to the following system of differential equations:
\[
\vec{x}'' = \begin{bmatrix} 1 & 3 \\ -3 & 5 \end{bmatrix} \vec{x} + \begin{bmatrix} -1 \\ 2 \end{bmatrix} \sin(3t).
\]
**Explanation:**
This problem involves finding a particular solution to a second-order vector differential equation. The given equation is expressed in matrix form:
- \(\vec{x}''\) represents the second derivative of the vector \(\vec{x}\) with respect to time \(t\).
- The coefficient matrix \(\begin{bmatrix} 1 & 3 \\ -3 & 5 \end{bmatrix}\) operates on the vector \(\vec{x}\).
- The term \(\begin{bmatrix} -1 \\ 2 \end{bmatrix} \sin(3t)\) is the non-homogeneous part, introducing a sinusoidal forcing function.
The goal is to determine the particular vector function \(\vec{x}(t)\) that satisfies this differential equation.
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