4 -9 x. -4 Find the most general real-valued solution to the linear system of differential equations 1 x1(t) -e^(-4t)cos(3t) e^(-4t)sin(3t) = C1 + c2 æ2(t) e^(-4t)(3sin(3t)) e^(-4t)(3cos(3t))
4 -9 x. -4 Find the most general real-valued solution to the linear system of differential equations 1 x1(t) -e^(-4t)cos(3t) e^(-4t)sin(3t) = C1 + c2 æ2(t) e^(-4t)(3sin(3t)) e^(-4t)(3cos(3t))
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:**
a. Find the most general real-valued solution to the linear system of differential equations given by:
\[
\dot{\vec{x}} =
\begin{bmatrix}
-4 & -9 \\
1 & -4
\end{bmatrix}
\vec{x}.
\]
**General Solution to the System:**
The solution can be expressed as:
\[
\begin{bmatrix}
x_1(t) \\
x_2(t)
\end{bmatrix}
=
c_1
\begin{bmatrix}
-e^{-4t} \cos(3t) \\
e^{-4t} (3 \sin(3t))
\end{bmatrix}
+
c_2
\begin{bmatrix}
e^{-4t} \sin(3t) \\
e^{-4t} (3 \cos(3t))
\end{bmatrix}
\]
where \(c_1\) and \(c_2\) are constants determined by initial conditions.
**Explanation of the Solution:**
- The system of differential equations is solved by finding the eigenvalues and eigenvectors of the matrix. The real parts of the eigenvalues contribute the exponential term \(e^{-4t}\), while the imaginary parts result in the sinusoidal functions (sine and cosine).
- The solution consists of two parts, each multiplied by an arbitrary constant \(c_1\) or \(c_2\), corresponding to the two linear independent solutions to the system.
- The presence of \(\cos(3t)\) and \(\sin(3t)\) indicates oscillatory behavior due to complex eigenvalues, modified by the exponential decay term \(e^{-4t}\).
This structure allows the system to model oscillations that decay over time, often seen in damped mechanical or electrical systems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb738ee4b-0d92-438d-b96f-d8d02fb325c0%2F60b667ac-5aac-4f09-8941-341182c83282%2Fev2nlpq_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
a. Find the most general real-valued solution to the linear system of differential equations given by:
\[
\dot{\vec{x}} =
\begin{bmatrix}
-4 & -9 \\
1 & -4
\end{bmatrix}
\vec{x}.
\]
**General Solution to the System:**
The solution can be expressed as:
\[
\begin{bmatrix}
x_1(t) \\
x_2(t)
\end{bmatrix}
=
c_1
\begin{bmatrix}
-e^{-4t} \cos(3t) \\
e^{-4t} (3 \sin(3t))
\end{bmatrix}
+
c_2
\begin{bmatrix}
e^{-4t} \sin(3t) \\
e^{-4t} (3 \cos(3t))
\end{bmatrix}
\]
where \(c_1\) and \(c_2\) are constants determined by initial conditions.
**Explanation of the Solution:**
- The system of differential equations is solved by finding the eigenvalues and eigenvectors of the matrix. The real parts of the eigenvalues contribute the exponential term \(e^{-4t}\), while the imaginary parts result in the sinusoidal functions (sine and cosine).
- The solution consists of two parts, each multiplied by an arbitrary constant \(c_1\) or \(c_2\), corresponding to the two linear independent solutions to the system.
- The presence of \(\cos(3t)\) and \(\sin(3t)\) indicates oscillatory behavior due to complex eigenvalues, modified by the exponential decay term \(e^{-4t}\).
This structure allows the system to model oscillations that decay over time, often seen in damped mechanical or electrical systems.
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