Chapter 7.6, Problem 3E

Interpretation Introduction

Interpretation:

The initial value problem is $\ddot{\text{x}}\text{ + x = ε}$ with $\text{x(0) = 1}$, $\dot{\text{x}}\text{(0) = 0}$

Solve the problem exactly

Using regular perturbation theory, find ${\text{x}}_{\text{0}}$, ${\text{x}}_{\text{1}}$, and ${\text{x}}_{\text{2}}$ in the series expansion ${\text{x(t,ε) = x}}_{\text{0}}{\text{(t) + εx}}_{\text{1}}{\text{(t) + ε}}^{\text{2}}{\text{x}}_{\text{2}}{\text{(t) + O(ε}}^{\text{3}}\text{)}$.

Explain why the perturbation solution does or doesn’t 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.