1. (a) What are A, x' and x'' in matrix form X'= AX of the system of differential equations ($) x₁ = 3x₁ + x2 { 2012 x2 # (b) Find all eigenvalues and eigenvectors of A. (c) Find exponential solutions of the system of differential equations. (d) Verify your solutions. = 5x1 - x2 ?
1. (a) What are A, x' and x'' in matrix form X'= AX of the system of differential equations ($) x₁ = 3x₁ + x2 { 2012 x2 # (b) Find all eigenvalues and eigenvectors of A. (c) Find exponential solutions of the system of differential equations. (d) Verify your solutions. = 5x1 - x2 ?
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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![**System of Differential Equations**
1. (a) What are \( \mathbf{A}, \mathbf{X} \) and \( \mathbf{X}' \) in matrix form \( \mathbf{X}' = \mathbf{A} \mathbf{X} \) for the system of differential equations:
\[
\begin{align*}
x_1' &= 3x_1 + x_2 \\
x_2' &= 5x_1 - x_2
\end{align*}
\]
(b) Find all eigenvalues and eigenvectors of \( \mathbf{A} \).
(c) Find exponential solutions of the system of differential equations.
(d) Verify your solutions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F801d34df-dcab-45aa-85e0-2349cce424a4%2Fd0cd6ecb-b2d4-4c06-b685-56a2adb6fdf8%2Fmgp6hre_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**System of Differential Equations**
1. (a) What are \( \mathbf{A}, \mathbf{X} \) and \( \mathbf{X}' \) in matrix form \( \mathbf{X}' = \mathbf{A} \mathbf{X} \) for the system of differential equations:
\[
\begin{align*}
x_1' &= 3x_1 + x_2 \\
x_2' &= 5x_1 - x_2
\end{align*}
\]
(b) Find all eigenvalues and eigenvectors of \( \mathbf{A} \).
(c) Find exponential solutions of the system of differential equations.
(d) Verify your solutions.
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