Chapter 3.5, Problem 3E

Interpretation Introduction

Interpretation:

To show that the equation $\frac{\text{d}\varphi }{\text{d}\tau }\text{ = f(}\varphi \text{)}$, where $\text{f(}\varphi \text{) = - sin }\varphi \text{ + γ}\sin \varphi \text{ cos }\varphi \text{ = sin }\varphi \text{ (γ cos }\varphi \text{ - 1)}$ reduces to $\frac{\text{d}\varphi }{\text{d}\tau }\text{ = A}\varphi \text{ - B}{\varphi }^{\text{3}}\text{ + O}\left({\varphi }^5\right)$. To find the values of the constants $\text{A}$ and $\text{B}$.

Concept Introduction:

The Taylor series expansion of the function $\text{f(x)}$ about the point $\text{x = a}$ is given as

${\text{f(x) = f(a) + f}}^{\text{'}}\left(\text{a}\right)\left(\text{x - a}\right)\text{ + }\frac{{\text{f}}^{\text{''}}\left(\text{a}\right)}{\text{2!}}{\left(\text{x - a}\right)}^{\text{2}}\text{+ }\mathrm{......................}$