Consider the system of differential equations Verify that z' = [( 1] ², 2(0) = | ¹0+ z(t) = c₁e¹t + c₂e 2¹ [¹] is a solution to the system of differential equations for any choice of the constants c₁ and c₂. Find values of c₁ and c₂ that solve the given initial value problem. (According to the uniqueness theorem, you have found the unique solution of ' =P*, (0) = o). z (t) = De]+( 4).e-2 [1]. help (numbers)
Consider the system of differential equations Verify that z' = [( 1] ², 2(0) = | ¹0+ z(t) = c₁e¹t + c₂e 2¹ [¹] is a solution to the system of differential equations for any choice of the constants c₁ and c₂. Find values of c₁ and c₂ that solve the given initial value problem. (According to the uniqueness theorem, you have found the unique solution of ' =P*, (0) = o). z (t) = De]+( 4).e-2 [1]. help (numbers)
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
4. Ordinary
![### System of Differential Equations
Consider the following system of differential equations:
\[
\vec{x}' = \begin{bmatrix} 1 & 3 \\ 3 & 1 \end{bmatrix} \vec{x}, \quad \vec{x}(0) = \begin{bmatrix} -12 \\ -2 \end{bmatrix}.
\]
### Verification Task
Verify that
\[
\vec{x}(t) = c_1 e^{4t} \begin{bmatrix} 1 \\ 1 \end{bmatrix} + c_2 e^{-2t} \begin{bmatrix} 1 \\ -1 \end{bmatrix}
\]
is a solution to the system of differential equations for any choice of the constants \( c_1 \) and \( c_2 \).
### Initial Value Problem
Find the values of \( c_1 \) and \( c_2 \) that solve the given initial value problem
with \(\vec{x} = P\vec{x}, \quad \vec{x}(0) = \vec{x}_0\).
According to the uniqueness theorem, you will find the unique solution.
\[
\vec{x}(t) = \left(\begin{bmatrix} \, \, \, \, \, \end{bmatrix} \right) \cdot e^{4t} \begin{bmatrix} 1 \\ 1 \end{bmatrix} + \left(\begin{bmatrix} \, \, \, \, \, \end{bmatrix} \right) \cdot e^{-2t} \begin{bmatrix} 1 \\ -1 \end{bmatrix}.
\]
For assistance, use the following [help (numbers)].
---
This text aims to guide learners through solving a system of differential equations using initial conditions to find specific constants. The solution involves determining \(c_1\) and \(c_2\) based on the initial vector and analyzing the exponential terms as projected onto the eigenvectors of the system matrix.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d6d6ec3-8d2a-4662-b20e-640089acaa34%2Ffcb352c0-3cef-414c-9633-3388d1f3e4ed%2Fdrgzo5h_processed.png&w=3840&q=75)
Transcribed Image Text:### System of Differential Equations
Consider the following system of differential equations:
\[
\vec{x}' = \begin{bmatrix} 1 & 3 \\ 3 & 1 \end{bmatrix} \vec{x}, \quad \vec{x}(0) = \begin{bmatrix} -12 \\ -2 \end{bmatrix}.
\]
### Verification Task
Verify that
\[
\vec{x}(t) = c_1 e^{4t} \begin{bmatrix} 1 \\ 1 \end{bmatrix} + c_2 e^{-2t} \begin{bmatrix} 1 \\ -1 \end{bmatrix}
\]
is a solution to the system of differential equations for any choice of the constants \( c_1 \) and \( c_2 \).
### Initial Value Problem
Find the values of \( c_1 \) and \( c_2 \) that solve the given initial value problem
with \(\vec{x} = P\vec{x}, \quad \vec{x}(0) = \vec{x}_0\).
According to the uniqueness theorem, you will find the unique solution.
\[
\vec{x}(t) = \left(\begin{bmatrix} \, \, \, \, \, \end{bmatrix} \right) \cdot e^{4t} \begin{bmatrix} 1 \\ 1 \end{bmatrix} + \left(\begin{bmatrix} \, \, \, \, \, \end{bmatrix} \right) \cdot e^{-2t} \begin{bmatrix} 1 \\ -1 \end{bmatrix}.
\]
For assistance, use the following [help (numbers)].
---
This text aims to guide learners through solving a system of differential equations using initial conditions to find specific constants. The solution involves determining \(c_1\) and \(c_2\) based on the initial vector and analyzing the exponential terms as projected onto the eigenvectors of the system matrix.
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