Chapter 7.6, Problem 1E

Interpretation Introduction

Interpretation:

To show that ${\text{x(t,ε) = (1- ε}}^{\text{2}}{\text{)}}^{\text{-1/2}}{\text{e}}^{\text{-εt}}\text{sin}\left[{\left({\text{1-ε}}^{\text{2}}\right)}^{\text{1/2}}\text{t}\right]$ is expanded as a power series in $\text{ε}$, we recover ${\text{x(t, ε) = (1- ε}}^{\text{2}}{\text{)}}^{\text{-1/2}}{\text{e}}^{\text{-εt}}\text{sin t - εt sin t}+{\rm O}\left({\text{ε}}^{\text{2}}\right)$.

Concept Introduction:

Usepower series expansion.

Answer to Problem 1E

Solution:

It is shown that for the given equation ${\text{x(t,ε) = (1- ε}}^{\text{2}}{\text{)}}^{\text{-1/2}}{\text{e}}^{\text{-εt}}\text{sin}\left[{\left({\text{1-ε}}^{\text{2}}\right)}^{\text{1/2}}\text{t}\right]$, power series expansion is $\text{x(t,}\tau \text{) = sin t - }\epsilon \text{t sin t}+{\rm O}\left({\text{ε}}^{\text{2}}\right)$.

Explanation of Solution

The power series expansion of the function ${\text{f(x)= a}}_{\text{n}}{\text{(x- c)}}^{\text{n}}$ is given as,

${\displaystyle \sum _0^{\infty }{\text{a}}_{\text{n}}{\left(\text{x - c}\right)}^{\text{n}}={\text{a}}_{\text{0}}\text{+}}{\text{ a}}_{\text{1}}\left(\text{x- c}\right){\text{+ a}}_{\text{2}}{\left(\text{x- c}\right)}^{\text{2}}\text{+}\mathrm{.....}$

Here, $\text{c}$ is the constant and ${\text{a}}_{\text{n}}$ the coefficient of the n^th term.

From the equation $\text{7}\text{.6}\text{.7}$

${\text{x(t,ε) = (1- ε}}^{\text{2}}{\text{)}}^{\text{-1/2}}{\text{e}}^{\text{-εt}}\text{sin}\left[{\left({\text{1-ε}}^{\text{2}}\right)}^{\text{1/2}}\text{t}\right]$

By differentiating the above equation with respect to $\text{ε}$,

$\frac{\text{dx(t,ε)}}{\text{dε}}\text{ = }\frac{\text{d}\left({{\text{(1- ε}}^{\text{2}}\text{)}}^{\text{-1/2}}{\text{e}}^{\text{-εt}}\text{sin}\left[{\left({\text{1-ε}}^{\text{2}}\right)}^{\text{1/2}}\text{t}\right]\right)}{\text{dε}}$

$\frac{\text{dx(t,ε)}}{\text{dε}}{\text{ = e}}^{\text{-εt}}\text{sin}\left[{\left({\text{1-ε}}^{\text{2}}\right)}^{\text{1/2}}\text{t}\right]\frac{\text{d}\left({{\text{(1- ε}}^{\text{2}}\text{)}}^{\text{-1/2}}\right)}{\text{dε}}+{{\text{(1- ε}}^{\text{2}}\text{)}}^{\text{-1/2}}\frac{\text{d}\left({\text{e}}^{\text{-εt}}\text{sin}\left[{\left({\text{1-ε}}^{\text{2}}\right)}^{\text{1/2}}\text{t}\right]\right)}{\text{dε}}$

$\frac{\text{dx(t,ε)}}{\text{dε}}{\text{ = e}}^{\text{-εt}}\text{sin}\left[{\left({\text{1-ε}}^{\text{2}}\right)}^{\text{1/2}}\text{t}\right]\left(\frac{-1}{2}{{\text{(1- ε}}^{\text{2}}\text{)}}^{\text{-3/2}}\left(-2\text{ε}\right)\right)+{{\text{(1- ε}}^{\text{2}}\text{)}}^{\text{-1/2}}\frac{\text{d}\left({\text{e}}^{\text{-εt}}\text{sin}\left[{\left({\text{1-ε}}^{\text{2}}\right)}^{\text{1/2}}\text{t}\right]\right)}{\text{dε}}$

$\frac{\text{dx(t,ε)}}{\text{dε}}{\text{ = e}}^{\text{-εt}}\text{sin}\left[{\left({\text{1-ε}}^{\text{2}}\right)}^{\text{1/2}}\text{t}\right]\left(\text{ε}\right){{\text{(1- ε}}^{\text{2}}\text{)}}^{\text{-3/2}}+{{\text{(1- ε}}^{\text{2}}\text{)}}^{\text{-1/2}}\frac{\text{d}\left({\text{e}}^{\text{-εt}}\text{sin}\left[{\left({\text{1-ε}}^{\text{2}}\right)}^{\text{1/2}}\text{t}\right]\right)}{\text{dε}}$

$\frac{\text{dx(t,ε)}}{\text{dε}}{\text{ = e}}^{\text{-εt}}\text{sin}\left[{\left({\text{1-ε}}^{\text{2}}\right)}^{\text{1/2}}\text{t}\right]\left(\text{ε}\right){{\text{(1- ε}}^{\text{2}}\text{)}}^{\text{-3/2}}+{{\text{(1- ε}}^{\text{2}}\text{)}}^{\text{-1/2}}{\text{e}}^{\text{-εt}}\frac{\text{d}\left(\text{sin}\left[{\left({\text{1-ε}}^{\text{2}}\right)}^{\text{1/2}}\text{t}\right]\right)}{\text{dε}}+{{\text{(1- ε}}^{\text{2}}\text{)}}^{\text{-1/2}}\text{sin}\left[{\left({\text{1-ε}}^{\text{2}}\right)}^{\text{1/2}}\text{t}\right]\frac{\text{d}\left({\text{e}}^{\text{-εt}}\right)}{\text{dε}}$

$\begin{array}{l}\frac{\text{dx(t,ε)}}{\text{dε}}{\text{ = e}}^{\text{-εt}}\text{sin}\left[{\left({\text{1-ε}}^{\text{2}}\right)}^{\text{1/2}}\text{t}\right]\left(\text{ε}\right){{\text{(1- ε}}^{\text{2}}}^{\text{)}}+{{\text{(1- ε}}^{\text{2}}}^{\text{)}}{\text{e}}^{\text{-εt}}\cos \left[{\left({\text{1- ε}}^{\text{2}}\right)}^{\text{1/2}}\text{t}\right]t\frac{1}{2}{\left({\text{1-ε}}^{\text{2}}\right)}^{\text{-1/2}}\left(-2\text{ε}\right)\\ {\text{ + (1- ε}}^{\text{2}}{\text{)}}^{\text{-1/2}}\text{sin}\left[{\left({\text{1-ε}}^{\text{2}}\right)}^{\text{1/2}}\text{t}\right]{\text{e}}^{\text{-εt}}\left(\text{-t}\right)\end{array}$

$\begin{array}{l}\frac{\text{dx(t,ε)}}{\text{dε}}{\text{ = e}}^{\text{-εt}}\text{sin}\left[{\left({\text{1-ε}}^{\text{2}}\right)}^{\text{1/2}}\text{t}\right]\left(\text{ε}\right){{\text{(1- ε}}^{\text{2}}}^{\text{)}}+{{\text{(1- ε}}^{\text{2}}}^{\text{)}}{\text{e}}^{\text{-εt}}\cos \left[{\left({\text{1- ε}}^{\text{2}}\right)}^{\text{1/2}}\text{t}\right]t{\left({\text{1-ε}}^{\text{2}}\right)}^{\text{-1/2}}\left(-\text{ε}\right)\\ {\text{ + (1- ε}}^{\text{2}}{\text{)}}^{\text{-1/2}}\text{sin}\left[{\left({\text{1-ε}}^{\text{2}}\right)}^{\text{1/2}}\text{t}\right]{\text{e}}^{\text{-εt}}\left(\text{-t}\right)\end{array}$

$\begin{array}{l}\frac{\text{dx(t,ε)}}{\text{dε}}{\text{ = e}}^{\text{-εt}}\text{sin}\left[{\left({\text{1-ε}}^{\text{2}}\right)}^{\text{1/2}}\text{t}\right]\left(\text{ε}\right){{\text{(1- ε}}^{\text{2}}}^{\text{)}}+{{\text{(1- ε}}^{\text{2}}}^{\text{)}}{\text{e}}^{\text{-εt}}\cos \left[{\left({\text{1- ε}}^{\text{2}}\right)}^{\text{1/2}}\text{t}\right]t{\left({\text{1-ε}}^{\text{2}}\right)}^{\text{-1/2}}\left(-\text{ε}\right)\\ {\text{ + (1- ε}}^{\text{2}}{\text{)}}^{\text{-1/2}}\text{sin}\left[{\left({\text{1-ε}}^{\text{2}}\right)}^{\text{1/2}}\text{t}\right]{\text{e}}^{\text{-εt}}\left(\text{-t}\right)\end{array}$

$\frac{\text{dx(t,ε)}}{\text{dε}}{\text{ = ε(1- ε}}^{\text{2}}{\text{)}}^{\text{-3/2}}{\text{e}}^{\text{-εt}}\text{sin}\left[{\left({\text{1-ε}}^{\text{2}}\right)}^{\text{1/2}}\text{t}\right]-{\text{εt(1- ε}}^{\text{2}}{\text{)}}^{\text{-1}}{\text{e}}^{\text{-εt}}\cos \left[{\left({\text{1- ε}}^{\text{2}}\right)}^{\text{1/2}}\text{t}\right]{\text{ - t(1- ε}}^{\text{2}}{\text{)}}^{\text{-1/2}}{\text{e}}^{\text{-εt}}\text{sin}\left[{\left({\text{1-ε}}^{\text{2}}\right)}^{\text{1/2}}\text{t}\right]$

The value of $\text{x(t,ε)}$ at $\text{ε}\to \text{0}$ is calculated as,

Substitute $\text{0}$ for $\text{ε}$ in the above expression $\text{x(t,ε)}$

$\text{x(t,0) = sin t}$

Value of $\frac{\text{dx(t,ε)}}{\text{dε}}$, as $\text{ε}\to \text{0}$

$\frac{\text{d}\left(\text{x(t, 0)}\right)}{\text{dε}}=\text{-t sin t}$

The power series expansion of $\text{x(t,ε)}$ is given as:

$\text{x(t,}\tau \text{)= }{\text{x(t,}\epsilon \text{)}|}_{\epsilon =0}+{\epsilon \frac{\text{dx(t,}\epsilon \text{)}}{d\epsilon }|}_{\epsilon =0}+{\rm O}\left({\text{ε}}^{\text{2}}\right)$

$\text{x(t,}\tau \text{) = x(t,0)}+\epsilon \frac{\text{dx(t, 0)}}{d\epsilon }+{\rm O}\left({\text{ε}}^{\text{2}}\right)$

$\text{x(t,}\tau \text{) = sin t}+\epsilon \left(\text{-t sin t}\right)+{\rm O}\left({\text{ε}}^{\text{2}}\right)$

$\text{x(t,}\tau \text{) = sin t - }\epsilon \text{t sin t}+{\rm O}\left({\text{ε}}^{\text{2}}\right)$

Hence, the power series expansion is $\text{x(t,}\tau \text{) = sin t - }\epsilon \text{t sin t}+{\rm O}\left({\text{ε}}^{\text{2}}\right)$ for the given equation ${\text{x(t,ε) = (1- ε}}^{\text{2}}{\text{)}}^{\text{-1/2}}{\text{e}}^{\text{-εt}}\text{sin}\left[{\left({\text{1-ε}}^{\text{2}}\right)}^{\text{1/2}}\text{t}\right]$.

Conclusion

It is shown that for the givenequation ${\text{x(t,ε) = (1- ε}}^{\text{2}}{\text{)}}^{\text{-1/2}}{\text{e}}^{\text{-εt}}\text{sin}\left[{\left({\text{1-ε}}^{\text{2}}\right)}^{\text{1/2}}\text{t}\right]$, the power series expansion is $\text{x(t,}\tau \text{) = sin t - }\epsilon \text{t sin t}+{\rm O}\left({\text{ε}}^{\text{2}}\right)$