Chapter 5.2, Problem 9E

Interpretation Introduction

Interpretation:

Find characteristic polynomial for system of linear equations, $\dot{\text{x}}\text{ = 4x - 3y}$ and $\dot{\text{y}}\text{ = 8x - 6y }$ using $\dot{\text{x}}\text{ = Ax}$ equation. Also find eigenvalues and eigenvectors of matrix A.

For the given system of linear equations, the general solution is found.

Classify the fixed points at the origin.

Concept Introduction:

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

The above linear system is 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 trace of matrix A, $\text{λ}$ is the corresponding eigenvalue, and $\text{Δ}$ is the determinant of matrix A.