Short Answer

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

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

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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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Fundamentals Of Differential Equations And Boundary Value Problems

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Short Answer

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

Step by Step Solution

Define Laplace transform

Find the value of Leatcosbt

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

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

Step by Step Solution

Define Laplace transform

Find the value of Leatcosbt

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