Chapter 7.6, Problem 24E

Interpretation Introduction

Interpretation:

The frequency $\text{ω}$ of periodic solutions of the system ${\ddot{\text{x}}\text{+x-εx}}^{\text{3}}\text{=0}$ is to be determined up to and including the $\text{O}\left({\text{ε}}^3\right)$ term.

Concept Introduction:

The Taylor series expansion of the function $\text{x}$ is

$\text{x}\left(\text{τ,t}\right){\text{=x}}_{\text{0}}\left(\text{τ}\right){\text{+εx}}_{\text{1}}\left(\text{τ}\right){\text{+ε}}^{\text{2}}{\text{x}}_{\text{2}}\left(\text{τ}\right)\text{+O}\left({\text{ε}}^{\text{3}}\right)$

The Taylor series expansion of the angular velocity function is

${\text{ω = 1+ εω}}_{\text{1}}{\text{+ε}}^2{\text{ω}}_2\text{+ O}\left({\text{ε}}^{\text{3}}\right)$

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

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