(c) dx dt -= 2x - 7y dy = 5x +10y + 4z dt dz z = 5y + 2z dt
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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Question
part B C
Transcribed Image Text:The image presents a system of differential equations labeled as (c). The equations are as follows:
1. \(\frac{dx}{dt} = 2x - 7y\)
2. \(\frac{dy}{dt} = 5x + 10y + 4z\)
3. \(\frac{dz}{dt} = 5y + 2z\)
These equations describe the rate of change of the variables \(x\), \(y\), and \(z\) with respect to time \(t\). The coefficients of each variable indicate their influence on the rates of change in the system.
![3. In the following problems, find the general solution of the given system.
(a)
\[
\frac{dx}{dt} = -\frac{5}{2}x + 2y
\]
\[
\frac{dy}{dt} = \frac{3}{4}x - 2y
\]
(b)
\[
X' =
\begin{pmatrix}
10 & -5 \\
8 & -12 \\
\end{pmatrix}
X
\]
This problem involves finding the general solution to systems of linear differential equations. In part (a), the system is expressed in terms of derivatives of \( x \) and \( y \) with respect to time \( t \). In part (b), the system is written in matrix form, where \( X \) is a vector function. Solving these systems typically involves techniques like eigenvalue and eigenvector analysis, or using matrix exponentials.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0a46b8d2-68f8-4ff5-85d9-0145753bf32f%2F18cd8eeb-a7f7-4f31-83c7-1a8eff594599%2Frhn2alk_processed.png&w=3840&q=75)
Transcribed Image Text:3. In the following problems, find the general solution of the given system.
(a)
\[
\frac{dx}{dt} = -\frac{5}{2}x + 2y
\]
\[
\frac{dy}{dt} = \frac{3}{4}x - 2y
\]
(b)
\[
X' =
\begin{pmatrix}
10 & -5 \\
8 & -12 \\
\end{pmatrix}
X
\]
This problem involves finding the general solution to systems of linear differential equations. In part (a), the system is expressed in terms of derivatives of \( x \) and \( y \) with respect to time \( t \). In part (b), the system is written in matrix form, where \( X \) is a vector function. Solving these systems typically involves techniques like eigenvalue and eigenvector analysis, or using matrix exponentials.
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