Q7.3 - 21E Given that L{cosbt}(s)=s/(s2+b2)... [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: Q7.3 - 21E (page 365)

Given that $L {cosbt} (s) = s / (s^{2} + b^{2})$ , use the translation property to compute $L {e^{at} cosbt}$ .

Short Answer

Expert verified

The value of $L \{e^{at} cosbt\}$ is $\frac{s - a}{{(s - a)}^{2} + b^{2}}$ .

Step by step solution

Define Laplace transform

When specific initial conditions are supplied, especially when the initial values are zero, the Laplace transform is a handy method of solving certain types of differential equations. Laplace transform $L$ of a function f(t) is defined as:

role="math" localid="1655792122645" $L {f (t)} = {鈭�}_{0}^{\frac{<}{>}} e^{- s t} f (t) d t$

In words, we can describe this expression as the Laplace transform of f(t) equals function F of s, that is, $L {f (t)} = F (s)$ .

Find the value of Leatcosbt

Given that $L {\cos b t} (s) = s / (s^{2} + b^{2})$ ,

Find $L \{e^{a t} \cos b t\} (s)$ using to translation property $L \{e^{a t} f (t)\} (s) = F (s - a)$ as:

$\begin{array}{rcl} L \{e^{a t} \cos b t\} (s) & = & F (s - a) \\ = & L {\cos b t} (s - a) \\ = & \frac{s - a}{{(s - a)}^{2} + b^{2}} \end{array}$

Hence, the value of $L \{e^{a t} \cos b t\}$ is $\frac{s - a}{{(s - a)}^{2} + b^{2}}$ .

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91影视

Given that $L {cosbt} (s) = s / (s^{2} + b^{2})$ , use the translation property to compute $L {e^{at} cosbt}$ .

Short Answer

Step by step solution

Define Laplace transform

Find the value of Leatcosbt

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91影视

Given that L{cosbt}(s)=s/(s2+b2), use the translation property to compute L{eatcosbt}.

Short Answer

Step by step solution

Define Laplace transform

Find the value of Leatcosbt

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Logic and Functions

Theoretical and Mathematical Physics

Mechanics Maths

Discrete Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

Given that $L {cosbt} (s) = s / (s^{2} + b^{2})$ , use the translation property to compute $L {e^{at} cosbt}$ .