Is a seL of functions 21,..., 2n such that the x,'s solve the equation and d = 0 for each i = 1,...,n. Find the equilibrium solutions to the systems below or show that there is no equilibrium solution. %3D (a) = 21 + 2x2 + 3x3 – 6 dr. = 2x1 – 3r2 + 2æ3 – 14 A = 3x1 + a2 – a'3 + 2 dra (b) 뿐3D 21-3x2 +7 dra = 2r - 6x2 – 7
Is a seL of functions 21,..., 2n such that the x,'s solve the equation and d = 0 for each i = 1,...,n. Find the equilibrium solutions to the systems below or show that there is no equilibrium solution. %3D (a) = 21 + 2x2 + 3x3 – 6 dr. = 2x1 – 3r2 + 2æ3 – 14 A = 3x1 + a2 – a'3 + 2 dra (b) 뿐3D 21-3x2 +7 dra = 2r - 6x2 – 7
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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![**Equilibrium Solutions for Systems of Equations**
**Problem 2**
An equilibrium solution for a system of equations is defined as a set of functions \(x_1, \ldots, x_n\) such that these functions solve the equations and \(\frac{dx_i}{dt} = 0\) for each \(i = 1, \ldots, n\).
Find the equilibrium solutions to the following systems or demonstrate that no equilibrium solution exists.
**System (a)**
\[
\begin{cases}
\frac{dx_1}{dt} = x_1 + 2x_2 + 3x_3 - 6 \\
\frac{dx_2}{dt} = 2x_1 - 3x_2 + 2x_3 - 14 \\
\frac{dx_3}{dt} = 3x_1 + x_2 - x_3 + 2
\end{cases}
\]
**System (b)**
\[
\begin{cases}
\frac{dx_1}{dt} = x_1 - 3x_2 + 7 \\
\frac{dx_2}{dt} = 2x_1 - 6x_2 - 7
\end{cases}
\]
**Objective**: For each system, determine if there are any equilibrium points by solving for \(x_1, x_2,\) and \(x_3\) (if applicable) where each derivative equals zero.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F85346d8e-a543-42db-b008-9fe5a873d3b5%2F2237e7a7-d03a-4cca-9a65-354ef97aeb81%2Finma4rr_processed.png&w=3840&q=75)
Transcribed Image Text:**Equilibrium Solutions for Systems of Equations**
**Problem 2**
An equilibrium solution for a system of equations is defined as a set of functions \(x_1, \ldots, x_n\) such that these functions solve the equations and \(\frac{dx_i}{dt} = 0\) for each \(i = 1, \ldots, n\).
Find the equilibrium solutions to the following systems or demonstrate that no equilibrium solution exists.
**System (a)**
\[
\begin{cases}
\frac{dx_1}{dt} = x_1 + 2x_2 + 3x_3 - 6 \\
\frac{dx_2}{dt} = 2x_1 - 3x_2 + 2x_3 - 14 \\
\frac{dx_3}{dt} = 3x_1 + x_2 - x_3 + 2
\end{cases}
\]
**System (b)**
\[
\begin{cases}
\frac{dx_1}{dt} = x_1 - 3x_2 + 7 \\
\frac{dx_2}{dt} = 2x_1 - 6x_2 - 7
\end{cases}
\]
**Objective**: For each system, determine if there are any equilibrium points by solving for \(x_1, x_2,\) and \(x_3\) (if applicable) where each derivative equals zero.
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