# Homework Help: Multiplication in a LaPlace Transform

1. May 25, 2010

### TG3

The problem statement, all variables and given/known data
Find the Laplace transform of the given function, a and b are real constants.
Hint: cos(bt)=(e^ibt+e^-ibt)/2 and sin(bt)=(e^it-e^ibt)/2i.

f(t)=e^(at)*sin(bt)

The attempt at a solution

The LaPlace transform of e^at is 1/(s-a).

The LaPlace transform of sin(bt) is b/(s^2+b^2).

Simply multiply those together, I got b/s^3-as^2+bs^2-ab^2.

This is wrong, and it feel like I'm making a very basic mistake that should be obvious, doing something other than multiplying the two seperate LaPlace transforms together. So, what am I supposed to do when the function I am supposed to work with has two easily distinguishable functions within it?

2. May 25, 2010

### LCKurtz

Have you had the shifting theorems? Like for L(eatf(t)) in terms of L(f(t))?

3. May 25, 2010

### TG3

I don't think so, but I'm not positive... I have trouble focusing for the entire period.

4. May 25, 2010

### LCKurtz

Well, if you haven't had the shifting theorem, you can always use the given hint. Express the sine in terms of exponentials, take the transform, and simplify it.

5. May 25, 2010

### vela

Staff Emeritus
Start by plugging that function into the definition of the Laplace transform and combining the exponentials. You might be able to see the answer immediately.