e influence...Find the Laplace transform of ecos(t), and esin(t).

Answered: e cos(t), and e sin(t).

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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e influence...
Find the Laplace transform of ecos(t), and esin(t).

Expert Solution

Step 1

Given:

$f (t) = e^{- t} \cos (t) a n d f (t) = e^{- t} \sin (t)$

Step 2

Explanation:

$f (t) = e^{- t} \cos (t) A p p l y t r a n s f o r m r u l e : I f F {f (t)} = F (s) t h e n L \{e^{a t} f (t)\} = F (s - a) F o r e^{- t} \cos (t) : f (t) = \cos (t), a = - 1 N o w u s e L {\cos a t} = \frac{s}{s^{2} + a^{2}} T h e r e f o r e, L {\cos t} = \frac{s}{s^{2} + 1^{2}} = \frac{s}{s^{2} + 1} T h e r e f o r e, L \{e^{a t} \cos t\} = F (s - (- 1)) L \{e^{a t} \cos t\} = F (s + 1) \Rightarrow L \{e^{a t} \cos t\} = \frac{s + 1}{{(s + 1)}^{2} + 1}$

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