Chapter 7.6, Problem 20E

Interpretation Introduction

Interpretation:

Using regular perturbation the solution of a nonlinear system $\ddot{\text{x}}\text{+}\text{x}\text{+}{\text{εx}}^3\text{=}\text{0}$ is $\text{x}\left(\text{t},\text{ε}\right)\text{ = }\text{a}\text{cos}\text{t}{\text{+ εa}}^{\text{3}}\left(\text{-}\frac{\text{3}}{\text{8}}\text{t}\text{sin}\text{t}\text{+ }\frac{\text{1}}{\text{32}}\left(\text{cos}\text{3t - cos}\text{t}\right)\right)+\text{O}\left({\text{ε}}^2\right)$ is to be shown and reason of its inferiority is to be explained.

Concept Introduction:

The solution of the equation of the form $\ddot{\text{x}}\text{+}\text{x}\text{+}\text{εh(x,}\dot{\text{x}}\text{)}\text{=}\text{0}$ is $\text{x}\left(\text{t,ε}\right){\text{=x}}_{\text{0}}\left(\text{t}\right){\text{+εx}}_{\text{1}}\left(\text{t}\right){\text{+ε}}^2{\text{x}}_2\left(\text{t}\right)\text{+}\mathrm{...}$

The expression of the second-order differential equation is,

$\frac{{\text{d}}^{\text{2}}\text{x}}{{\text{dt}}^{\text{2}}}\text{+}\text{P}\frac{\text{dx}}{\text{dt}}\text{+}\text{Q}\text{=}\text{0}$

Where, $\text{P}$ and $\text{Q}$ are constants.

The solution of this second-order differential equation is,

${\text{x(t)=c}}_{\text{1}}{\text{e}}^{{\text{m}}_{\text{1}}\text{t}}{\text{+c}}_{\text{2}}{\text{e}}^{{\text{m}}_{\text{2}}\text{t}}$