Chapter 5.2, Problem 5E

Interpretation Introduction

Interpretation:

Find the characteristic polynomial for the system of linear equations $\dot{\text{x}}\text{ = 3x- 4y}$ and $\dot{\text{y}}\text{ = x -y}$ using $\dot{\text{x}}\text{ = Ax}$ equation. Also, determine the eigenvalues and eigenvectors of matrix A.

To find the general solution for the given system of linear equations.

Classify the fixed points at the origin.

Concept Introduction:

For two dimensional linear systems, equations are $\dot{\text{x}}\text{ = ax + by}$, $\dot{\text{y}}\text{ = cx + dy}$

Above linear system expressed in the form $\dot{\text{x}}\text{ = Ax}$.

The standard characteristics polynomials is,

${\text{λ}}^{\text{2}}\text{- τλ + Δ = 0}$, where $\text{τ}$ is the trace of matrix A, $\text{λ}$ is the corresponding eigenvalue, and $\text{Δ}$ is determinant of matrix A.