Another Laplace Transform Question

[jumbogala]

Homework Statement

Find the Laplace transform of f(t).

f(t) = -9t + 7 + u₅(-7 + 9t + e^t)

Homework Equations

The Attempt at a Solution

The Laplace transform of -9t is just -9/t²

The Laplace transform of 7 is 7/s

But I am not sure how to find it for u₅(-7 + 9t + e^t). In the tables it says u_a(t)f(t-a) = e^-asF(s). Is that the line I should use? Even if it is, I don't know how.

 

[tjackson3]

You've got the right start. You want to take it term by term, and you have the Laplace transform down for the first two terms. The way to handle step functions is as follows. First, get rid of the step function by saying:

$$L[u_c(f(t))] = e^{-cs}L[f(t+c)]$$

So in your case, the third term would become

$$e^{-5s} L[-2 + 9t + e^t]$$

Now it's just another term by term Laplace transform. So your final answer should be

$$-9/s^2 + 7/s + e^{5s}(-2/s + 9/s^2 + 1/s)$$

n.b.: Remember that the Laplace transform is always a function of s, while the inverse Laplace transform is a function of t. I see you got it slightly mixed up, so just something to keep in mind! :)