Chapter 7.6, Problem 26E

Interpretation Introduction

Interpretation:

To find the exact solution to the equation ${\dot{\text{x}}\text{ = -ε x sin}}^{\text{2}}\text{t}$, with $0\le \text{ε << 1}$ and ${\text{x = x}}_{\text{0}}$, at $\text{t = 0}$. To show that $\text{x}\left(\text{t}\right)\text{ = }\overline{\text{x}}\left(\text{t}\right)\text{ + O}\left(\text{ε}\right)$, if $\overline{\text{x}}\left(\text{t}\right)\text{ = }\frac{\text{1}}{\text{2π}}{\displaystyle \underset{\text{t - π}}{\overset{\text{t + π}}{\int }}\text{x}\left(\text{τ}\right)\text{dτ}}$. To find an approximate differential equation satisfied by $\overline{\text{x}}$, using the method of averaging and to solve it. To compare both the results and find how large the error incurred by averaging is.

Concept Introduction:

• ➢ The exact solution can be found by separation of variables.

• ➢ Leibniz’s integral is given by

  $\frac{\text{d}}{\text{dt}}\left(\frac{\text{1}}{\text{2π}}{\displaystyle \underset{\text{t - π}}{\overset{\text{t + π}}{\int }}\text{x}\left(\text{τ}\right)\text{dτ}}\right)\text{ = }\frac{\text{1}}{\text{2π}}\left(\text{ x}\left(\text{t + π}\right)\left(\frac{\text{d}\left(\text{t + π}\right)}{\text{dt}}\right)\text{- x}\left(\text{t - π}\right)\left(\frac{\text{d}\left(\text{t - π}\right)}{\text{dt}}\right)+{\displaystyle \underset{\text{t - π}}{\overset{\text{t + π}}{\int }}\frac{\partial }{\partial \text{t}}\text{x}\left(\text{τ}\right)\text{dτ}}\right)$