Chapter 2.4, Problem 9E

Interpretation Introduction

Interpretation:

The analytical solution to ${\dot{\text{x}}\text{ = -x}}^{\text{3}}$ for an arbitrary initial condition and the proof of $\text{x(t)}\to \text{0 as t}\to \infty \text{, }$ but the decay is not exponential is to be obtained. Also, a numerical accurate graph for the initial condition ${\text{x}}_0\text{ = 10, for 0}\le \text{t}\le \text{10 }$ is to be plotted including a solution to $\dot{\text{x}}\text{ = -x}$ for the same initial condition.

Concept Introduction:

In the second-order phase transition. The system comes to an equilibrium so much slower than usualwhich is known as ‘critical slowing down’. One of such a transition is represented by a system ${\dot{\text{x}}\text{ = -x}}^{\text{3}}$, but this decay is not an exponential decay. $\dot{\text{x}}\text{ = -x}$ shows the exponential decay.