Chapter 5.2, Problem 12E

Interpretation Introduction

Interpretation:

Write the system equation in two-dimensional linear system. Show that for $R>0$, the origin is asymptotically stable and it is neutrally stable if $R=0$. For $R^{2}C-4L$ positive, negative, or zero, classify the fixed point at the origin, and plot the phase diagram.

Concept Introduction:

A two dimensional linear system can be expressed in the form as

$\begin{array}{l}\dot{\text{x}}\text{=ax+by}\\ \dot{\text{y}}\text{=cx+dy}\\ \end{array}$

Here a, b, c, d are parameters. This linear system can be represented in the matrix form as

$\dot{\text{x}}\text{=Ax}$

Here the matrix $\dot{\text{x}}$ is represented as

$\dot{\text{x}}\text{ = }\left[\begin{array}{c}\dot{\text{x}}\\ \dot{\text{y}}\end{array}\right]$ and $\text{A=}\left(\begin{array}{cc}\text{a}& \text{b}\\ \text{c}& \text{d}\end{array}\right)$

$\left[\begin{array}{c}\dot{\text{x}}\\ \dot{\text{y}}\end{array}\right]\text{=}\left(\begin{array}{cc}\text{a}& \text{b}\\ \text{c}& \text{d}\end{array}\right)\left(\begin{array}{c}\text{x}\\ \text{y}\end{array}\right)$