Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]
$\cos^{3} t$

The Laplace transform for the given equation is $\frac{s^{3} + 7 s}{(s^{2} + 9) (s^{2} + 1)}$ .

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TABLE OF CONTENTS

Definition of Laplace transform

The integral transform of a given derivative function with real variable t into a complex function with variable s is known as the Laplace transform.
Let f(t) be supplied for t(0), and assume that the function meets certain constraints that will be presented subsequently.
The Laplace transform formula defines the Laplace transform of f(t), which is indicated by $L \{f (t)\}$ or F(s).

Determine the Laplace transform for the given equation

Given that $\cos^{3} t$ ,

Find the Laplace transform of $\cos^{3} t$ using $c o s^{3} a = \frac{1}{4} (c o s 3 a + 3 c o s a)$ , $L {a f (x) \pm b g (x)} = a L {f} \pm b L {g (t)}$ , $L {c o s b t} = \frac{s}{s^{2} + b^{2}}$ and $\frac{a}{b} \pm \frac{c}{d} = \frac{d a \pm b c}{b d}$ as:

$\begin{array}{rcl} L \{\cos^{3} t\} & = & L \{\frac{1}{4} (\cos 3 t + 3 \cos t)\} \\ = & \frac{1}{4} L {\cos 3 t + 3 \cos t} \\ = & \frac{1}{4} [L {\cos 3 t} + 3 L {\cos t}] \\ = & \frac{1}{4} [\frac{s}{s^{2} + 3^{2}} + 3 \times \frac{s}{s^{2} + 1^{2}}] \end{array}$

Simplify the equation as:

$\begin{array}{rcl} L \{\cos^{3} t\} & = & \frac{1}{4} [\frac{s}{s^{2} + 9} + \frac{3 s}{s^{2} + 1}] \\ = & \frac{1}{4} [\frac{s (s^{2} + 1) + 3 s (s^{2} + 9)}{(s^{2} + 9) (s^{2} + 1)}] \\ = & \frac{1}{4} [\frac{s^{3} + s + 3 s^{3} + 27 s}{(s^{2} + 9) (s^{2} + 1)}] \\ = & \frac{1}{4} [\frac{{\overset{⏞}{4 s^{3} + 28 s}}^{4 Common}}{(s^{2} + 9) (s^{2} + 1)}] \end{array}$

Further simplifying the equation as follows:

$\begin{array}{rcl} L \{\cos^{3} t\} & = & 4 [\frac{4 (s^{3} + 7 s)}{(s^{2} + 9) (s^{2} + 1)}] \\ = & \frac{s^{3} + 7 s}{(s^{2} + 9) (s^{2} + 1)} \end{array}$

Therefore, the Laplace transform for the given equation is $\frac{s^{3} + 7 s}{(s^{2} + 9) (s^{2} + 1)}$ .

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Fundamentals Of Differential Equations And Boundary Value Problems

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Short Answer

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]
$\cos^{3} t$

Step by Step Solution

Definition of Laplace transform

Determine the Laplace transform for the given equation

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Fundamentals Of Differential Equations And Boundary Value Problems

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Short Answer

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]cos3t

Step by Step Solution

Definition of Laplace transform

Determine the Laplace transform for the given equation

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In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]
$\cos^{3} t$