Chapter 5.2, Problem 8E

Interpretation Introduction

Interpretation:

Find the characteristic polynomial for the system of linear equations $\dot{\text{x}}\text{ = -3x + 4y}$ and $\dot{\text{y}}\text{ = -2x + 3y }$ using $\dot{\text{x}}\text{ = Ax}$ equation. The eigenvalues and the eigenvectors of the matrix A is to be found.

Solve the given system of linear equations and write the general solution.

The fixed points at the origin is to be classified.

Concept Introduction:

The two dimensional linear system 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 the determinant of matrix A.