Chapter 5.2, Problem 4E

Interpretation Introduction

Interpretation:

Find characteristic polynomial for the system of linear equations, $\dot{\text{x}}\text{ = 5x+10y}$ and $\dot{\text{y}}\text{ = -x -y}$ using $\dot{\text{x}}\text{ = Ax}$ equation. Also find eigenvalues and eigenvectors.

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

Classify the fixed points at the origin.

Concept Introduction:

Equations for two dimensional linear system 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 characteristic polynomial is

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