00:01
All right, so here we want to take the laplace transform of this fraction.
00:06
It looks a little bit scary, but we have tricks on to solve this.
00:10
First, you might check to see if you can factor, because then we could get partial fractions, but it doesn't really look good for factoring.
00:18
So our next trick is to go ahead and complete the square.
00:23
So on the bottom, we are going to take half of the middle term and set it up as s plus 4 squared.
00:30
And then we're going to check to make sure we haven't changed anything.
00:33
So in a honor cell or to our cells, we think, okay, if i boil it out, i get s squared plus 8 s plus 16.
00:43
But i have actually 20.
00:44
So i better add four more.
00:46
And then these are equivalent.
00:48
Okay.
00:49
So now we are having to do one more trick before we can take the inverse laplace transform.
00:58
And that is when we have this.
01:00
Shift in our power term here, we need to make sure that our s on top has the same shift.
01:08
So if i'm going to create that shift on this first term, then i have to compensate by subtracting it and then when i combine it, i'm still the same as before.
01:22
But now that i've done this form, then it's really useful because now i can think backwards.
01:27
So because i have s plus 4 and s plus 4, that shift in the frequency domain is an exponential multiplier in the time domain.
01:40
So i get e to the minus 4t times.
01:42
And otherwise, it's in the form of cosine square root of this.
01:46
So i get cosine 2t...