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" $L {f (t)} = \int_{0}^{\frac{<}{>}} e^{- s t} f (t) d t$

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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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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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

Theoretical and Mathematical Physics

Calculus

Decision Maths

Applied Mathematics

Pure Maths

Geometry

Study anywhere. Anytime. Across all devices.

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