00:01
All right, we're going to try to find the laplace transform of t squared sine of 2t.
00:07
Now, the general form for solving laplace transfer, when you have a polynomial multiplier, it looks a little scary.
00:17
So let's kind of show you what that looks like.
00:19
If i have t squared and then some function of t, then i'm going to get minus one.
00:25
In this case, because i'm squared, we'll actually get a positive because i can take the minus one and square it.
00:31
But then i get a second derivative with respect to s of my f of s, the laplace of that.
00:40
So i've got a bit of work to do to solve this.
00:44
Really, it's just the second derivative because the minus one is squared of the laplace of the function.
00:51
So i went ahead and wrote down for you here the laplace transform.
00:56
You can find this right off the table for the function, whereas this will be our, f of t.
01:03
So you can see that the laplace transform is really going to be nothing more than the second derivative of f of s.
01:10
So let's find that.
01:13
Find the derivative and then we'll be basically done with this problem.
01:17
Okay, so let's start by finding the first derivative.
01:20
And i'm going to rewrite the function so i can do some successive power rule type things.
01:27
Okay, so i'm going to rewrite it as two s squared plus four of the minus one.
01:31
So when i find the derivative, then i will do power rule.
01:35
So i'll get minus two, s squared plus four to the minus two power times derivative of the inside, which is two s.
01:44
And i can clean that up to be minus four s, s squared plus four to the minus two.
01:51
All right, we've got one more derivative to do.
01:54
So we're actually almost done.
01:55
It really wasn't too bad.
01:56
We just have to do this lovely derivative.
01:59
Okay, and this is a product.
02:00
So we're going to have to do product rule.
02:04
Okay, so let's do derivative of the first.
02:07
So i get minus four times the second.
02:10
We're doing the derivative with respect to x.
02:14
Then my pen is acting weird.
02:16
Okay.
02:17
And then we're going to add to it the derivative of the second times the first.
02:22
A little easier if i just write the first because it's cleaner.
02:25
Now i'm going to multiply by the derivative of the second.
02:28
So i'll get minus two.
02:31
Times s squared plus four to the minus three times yet another two s due to chain rolls.
02:38
Let's see what we have here.
02:40
So it looks like we have minus four over s squared plus four squared.
02:47
And i've got a lot of stuff here.
02:49
It looks like a positive 16 s squared over s squared plus four cubed.
02:59
I can combine this and i feel like that will give me.
03:13
Okay, so to combine it, i need to get a common denominator, i'm going to multiply the the left term.
03:21
I'll kind of squeeze it in here.
03:22
I'm going to multiply the left term by s squared plus four over s squared plus four.
03:28
That will give me a common denominator.
03:29
So now the big numerator is minus 4 s squared.
03:33
I'm going to distribute that minus 4, minus 16 plus 16 s squared all over.
03:40
Now we have s squared plus 4 cubed, but at least i made it one fraction.
03:44
We're almost there.
03:45
I'm going to continue it over here on the left.
03:50
So basically we will get, we'll clean it up, and it looks like we get 12 s squared minus 16 all over s squared plus four cubed.
04:03
That is our second derivative, which we know is also the laplace transform of what we want.
04:07
Let's write it now.
04:09
Laplace transform.
04:10
That's a funny one.
04:12
Let's fix that.
04:14
Let's get that fixed.
04:16
I'll make it look a little nicer.
04:18
So the laplace transform of t squared sine of 2t is 12 s squared minus 16 all over s squared plus four cute.
04:30
So we did it.
04:32
Success.
04:33
Okay, but we're only just beginning.
04:35
We've got a few more things to do.
04:36
So i'm going to stop and clear the board, and we'll continue on with the next one.
04:44
All right.
04:44
Our next thing to do is to find the inverse laplace transform of this.
04:50
And it's not quite in the right form.
04:52
There's usually when it looks like this.
04:53
We have two choices, typically.
04:55
One is to see if we can factor, and the second is to complete the square, and factoring isn't looking very doable.
05:02
So let's go ahead and complete the square.
05:06
We need to do that to get in a form where we can think backwards or use their laplace, inverse laplace transfer in tables.
05:14
Okay, so how i complete the square is i look at the middle term.
05:18
I take half to get my number, s minus two.
05:22
If i square it, notice that that naturally would be s squared minus 4 s plus 4, but i really need a total of five.
05:30
So i'm going to complete it by adding one.
05:32
And now the two expressions are equivalent.
05:35
All right, excellent.
05:37
Okay, so now that i have this form, i'm actually good to go.
05:40
I can already think backwards and finish this off.
05:43
This is of the form for sign of t.
05:46
This shift tells me i have an exponential multiplier.
05:50
So my answer is e to the 2t, sign of t.
05:56
Okay, excellent.
05:58
Hopefully that helped...