In 1–20, determine the Laplace transform of | Ch 7.3 - 19E

Fundamentals of Differential Equations | 8th Edition | ISBN: 9780321747730 | Authors: R. Kent Nagle, Edward B. Saff, Arthur David Snider

ISBN: 9780321747730 43

Solution for problem 19E Chapter 7.3

Fundamentals of Differential Equations | 8th Edition

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Fundamentals of Differential Equations | 8th Edition | ISBN: 9780321747730 | Authors: R. Kent Nagle, Edward B. Saff, Arthur David Snider

Fundamentals of Differential Equations | 8th Edition

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Problem 19E

In Problems 1–20, determine the Laplace transform of the given function using Table 7.1 and the properties of the transform given in Table 7.2. [Hint: In Problems 12–20, use an appropriate trigonometric identity.]

\(\text { cosnt }\) \(\text { sinmt }\)

\(m \neq n\)

Equation transcription:

$cosnt$

$sinmt$

$m\ne{}n$

Text transcription:

{ cosnt }

{ sinmt }

m neq n

Step-by-Step Solution:

Solution

Here,

$f(t)=cos\ nt\ sin\ mt\$

where, $n\ne{}m\$

We have to find $L[f(t)]=?$

Step 1

We know that as per basic trigonometric expansion

$cos\ x\ sin\ y\ =\frac{1}{2}[sin(x+y)-sin(x-y)]$

So,

$cos\ nt\ sin\ mt\ =\frac{1}{2}[sin(n+m)t-sin(n-m)t]$

According to laplace transformation

$L(cos\ nt\ sin\ mt)\ =\frac{1}{2}[\frac{n+m}{{s}^{2}+(n+m{)}^{2}}+\frac{n-m}{{s}^{2}+(n-m{)}^{2}}]$

On solving the expansion, we get

$L(cos\ nt\ sin\ mt)\ =\frac{1}{2}[\frac{2m{s}^{2}-({n}^{2}-{m}^{2})\times{}2m}{[{s}^{2}+(n+m{)}^{2}][{s}^{2}+(n-m{)}^{2}]}]$

Therefore,

$L(cos\ nt\ sin\ mt)\ =\frac{m({m}^{2}-{n}^{2}+{s}^{2})}{[{s}^{2}+(n+m{)}^{2}][{s}^{2}+(n-m{)}^{2}]}$ .

Step 2 of 1

Chapter 7.3, Problem 19E is Solved

Textbook: Fundamentals of Differential Equations

Edition: 8

Author: R. Kent Nagle, Edward B. Saff, Arthur David Snider

ISBN: 9780321747730

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