Q16RP In Problems 3-10, determine the ... [FREE SOLUTION]

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Fundamentals Of Differential Equations And Boundary Value Problems

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

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 7: Q16RP (page 416)

In Problems 3-10, determine the Laplace transform of the given function. $\frac{1}{{(s^{2} + 9)}^{2}}$

Short Answer

Expert verified

Therefore, the solution is $L^{鈭� 1} {\frac{1}{{(s^{2} + 9)}^{2}}} = \frac{\sin 3 t 鈭� 3 t \cos 3 t}{54}$

Step by step solution

Given Information

The given function value in s domain is $\frac{1}{{(s^{2} + 9)}^{2}}$ .

Use convolution theorem

Let $f (t) = \sin 3 t$ Then

$\begin{matrix} F (s) = L^{鈭� 1} {f (t)} (s) \\ = L^{鈭� 1} {\sin 3 t} (s) \\ = \frac{3}{s^{2} + 3^{2}} \end{matrix}$

We can write the given function as

$\frac{1}{{(s^{2} + 9)}^{2}} = \frac{1}{9} \frac{3}{s^{2} + 3^{2}} \frac{3}{s^{2} + 3^{2}}$

Now, using convolution theorem we get

$\begin{matrix} L^{鈭� 1} \{\frac{1}{{(s^{2} + 9)}^{2}}\} = L^{鈭� 1} \{\frac{1}{9} \frac{3}{s^{2} + 3^{2}} \frac{3}{s^{2} + 3^{2}}\} \\ = \frac{1}{9} L^{鈭� 1} {F (s) F (s)} \\ = \frac{1}{9} [f (t) * f (t)] \\ = \frac{1}{9} [\sin 3 t * \sin 3 t] \end{matrix}$

Simplify further as:

$\begin{matrix} L^{鈭� 1} \{\frac{1}{{(s^{2} + 9)}^{2}}\} = \frac{1}{9} {鈭�}_{0}^{t} \sin 3 (t 鈭� v) \sin 3 t d v \\ = \frac{1}{18} {鈭�}_{0}^{t} [\cos (6 v 鈭� 3 t) 鈭� \cos 3 t] d v \\ = \frac{1}{18} ({鈭�}_{0}^{t} \cos (6 v 鈭� 3 t) d v 鈭� {鈭�}_{0}^{t} \cos 3 t d v) \\ = \frac{1}{18} ({[\frac{\sin (6 v 鈭� 3 t)}{6}]}_{0}^{t} 鈭� \cos 3 t {[v]}_{0}^{t}) \\ = \frac{1}{18} (\frac{\sin 3 t}{6} + \frac{\sin 3 t}{6} 鈭� t \cos 3 t) \end{matrix}$

Further simplify as:

$\begin{matrix} L^{鈭� 1} \{\frac{1}{{(s^{2} + 9)}^{2}}\} = \frac{1}{18} \frac{\sin 3 t 鈭� 3 t \cos 3 t}{3} \\ = \frac{\sin 3 t 鈭� 3 t \cos 3 t}{54} \end{matrix}$

Therefore, the required inverse Laplace transform is

$L^{鈭� 1} \{\frac{1}{{(s^{2} + 9)}^{2}}\} = \frac{\sin 3 t 鈭� 3 t \cos 3 t}{54}$

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91影视

In Problems 3-10, determine the Laplace transform of the given function. $\frac{1}{{(s^{2} + 9)}^{2}}$

Short Answer

Step by step solution

Given Information

Use convolution theorem

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91影视

In Problems 3-10, determine the Laplace transform of the given function.1(s2+9)2

Short Answer

Step by step solution

Given Information

Use convolution theorem

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Most popular questions from this chapter

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Probability and Statistics

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In Problems 3-10, determine the Laplace transform of the given function. $\frac{1}{{(s^{2} + 9)}^{2}}$