Chapter 7.6, Problem 2E

Interpretation Introduction

Interpretation:

To obtain the exact solution to the initial value problem $\ddot{\text{x}}\text{ + }\dot{\text{x}}\text{ + εx = 0, with x}\left(0\right)=0$. Using regular perturbation theory, find $\text{x0}$, $\text{x1}$, and $\text{x2}$ in the series expansion $\text{x}\left(\text{t,ε}\right){\text{=x}}_{\text{0}}{\text{+ εx}}_{\text{1}}{\text{+ εx}}_{\text{2}}\text{+ O}\left({\text{ε}}^{\text{3}}\right)$. Does the perturbation solution contain secular terms?

Concept Introduction:

Finding complimentary function and particular integral, we can find the total/complete solution of the given differential equation. Then using initial conditions, we can find the complete solution of the given second-order differential equation.

From the regular perturbation theorem, the function ${\text{x(t,ε) = x}}_{\text{0}}{\text{(t,ε) + εx}}_{\text{1}}{\text{(t,ε) + ε}}^{\text{2}}{\text{x}}_{\text{2}}{\text{(t,ε) + O(ε}}^{\text{3}}\text{)}$.

Here, $\text{ε}$ is constant.