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RoshanBBQ
RoshanBBQ is offline
#3
Apr4-12, 04:15 AM
P: 280
Quote Quote by NewtonianAlch View Post
1. The problem statement, all variables and given/known data
Determine the Laplace transform:

g(t) = 2*e[itex]^{-4t}[/itex]u(t-1)

3. The attempt at a solution

Essentially we're told for a time shift we multiply the Laplace transform pair of the function (without the delay) by e[itex]^{-as}[/itex]

So here a = 1 (for the delay)

The Laplace transform for e[itex]^{-4t}[/itex] is [itex]\frac{1}{s + 4}[/itex]

Multiplying we should get e[itex]^{-s}[/itex]([itex]\frac{2}{s + 4}[/itex])

However the answer is e[itex]^{-s}[/itex]*([itex]\frac{2}{s + 4}[/itex]) * [itex]\frac{1}{e^{4}}[/itex]
It's because you must be in the form
[tex]e^{-4(t-a)}u(t-a)[/tex]
Take a look at your exponential. It isn't time-shifted by a to the right. What you can do is use the fact that exponentials multiplied add their exponents and the fact that e^b/e^b = 1, so you can multiply by it without changing your values. So you choose e^b so that you can add its exponents and arrive to the needed shift. The e^b in the denominator is factored outside of the linear inverse Laplace operator and you go from there.