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- Now I think is a good time to add some notation and
- techniques to our Laplace Transform tool kit.
- So the first thing I want to introduce is just kind of a
- quick way of doing something.
- And that is, if I had the Laplace Transform, let's say I
- want to take the Laplace Transform of the second
- derivative of y.
- Well, we proved several videos ago that if I wanted to take
- the Laplace Transform of the first derivative of y, that is
- equal to s times the Laplace Transform of y minus y of 0.
- And we used this property in the last couple of videos to
- actually figure out the Laplace Transform of the
- second derivative.
- Because if you just, you know, if you say this is y prime,
- this is the anti-derivative of it, then you could just
- pattern match.
- You could say, well, the Laplace Transform of y prime
- prime, that's just equal to s times the Laplace Transform of
- y prime minus y prime of 0.
- This is the derivative of this, just like this is the
- derivative of this.
- I'll draw a line here just so you don't get confused.
- So the Laplace Transform of y prime prime is this thing.
- And now we can use this, which we proved several videos ago,
- to resubstitute it and get it in terms of the Laplace
- Transform of y.
- So we can expand this part.
- The Laplace Transform of the derivative of y, that's just
- equal to s times the Laplace Transform of y minus y of 0.
- And then we have the outside, right?
- We have s minus y prime of 0.
- And then when you expand it all out, and we've done this
- before, you get s squared times the Laplace Transform of
- y minus s times y of 0 minus y prime of 0.
- Now there's something interesting to note here, and
- if you learn this it'll make it a lot faster.
- You won't have to go through all of this and risk making
- careless mistakes when you have scarce time and paper on
- your test. Just notice that when you take the Laplace
- Transform of the second derivative, what do we end up?
- We end up with s squared, right?
- This was the second derivative.
- So I end up with s squared times the Laplace Transform of
- y, minus s times y of 0 minus 1 times y prime of 0.
- So every term, we started with s squared, and then every term
- we lower the degree of s one, and then everything except the
- first term is a negative sign.
- And then we started with the Laplace Transform of y, and
- then you can almost view the Laplace Transform as a kind of
- integral, so we kind of take the derivatives, so
- then you get y.
- And then you take the derivative
- again, you get y prime.
- And of course every other term is negative.
- And these aren't the actual functions.
- These are those functions evaluated at 0.
- But that's a good way to help you, hopefully, remember how
- to do these.
- And once you get the hang of it, you can take the Laplace
- Transform of any arbitrary function very, very quickly.
- Or any arbitrary derivatives.
- So let's say we wanted to take the Laplace Transform of, I
- don't know, this should hit the point home, the fourth
- derivative of y.
- That 4 in parentheses means the fourth derivative.
- I could have drawn four prime marks, but either way.
- So what is this equal to?
- If we use this technique and substitute it, we're bound to
- make some form of careless mistake or other, and it would
- take us forever and it would waste a lot of paper.
- But now we see the pattern, and so we can just say, well,
- the Laplace Transform of this, in terms of the Laplace
- Transform of y, right, that's what we want to get to, is
- going to be s to the fourth times the Laplace Transform of
- y-- now every other term is going to have a minus in front
- of it-- minus-- lower the degree on the s--
- minus s to the third.
- And then you could kind of say, let's take the, you know,
- so form of derivative, so you get y of 0-- it's not a real
- derivative.
- The Laplace Transform really isn't the anti-derivative of y
- of 0, but anyway, I think you get the idea.
- And then we lower the degree on s again, minus s squared,
- take the derivative.
- And of course these aren't functions.
- But we're evaluating the derivative of that
- function now of 0.
- So y prime of 0, minus-- now we lower the degree one more--
- minus s, times-- this is an s-- times y prime prime of 0.
- We have one more term.
- Lower the degree on the s one more time.
- Then you get s to the 0, which is just 1.
- So minus-- and 1 is a coefficient-- and then you
- have y, the third derivative of y-- let me scroll over a
- little bit-- the third derivative of y
- evaluated at 0.
- So I think you see the pattern now.
- And this is a much faster way of evaluating the Laplace
- Transform of an arbitrary derivative of y, as opposed to
- keep going through that pattern over and over again.
- Another thing I want to introduce you to is just a
- notational savings.
- And it's just something that you'll see, so you might as
- well get used to it.
- And it actually saves time over, you know, keep writing
- this curly L in this bracket.
- If I have the Laplace Transform of y of t, I can
- write as, and people tend to write it as-- well, it's going
- to be a function of s, and what they use is a capital Y
- to denote the function of s.
- It makes sense, because normally when we're doing
- antiderivatives, you just take-- you know, when you
- learn the fundamental theorem of calculus, you learn that
- the integral of f with respect to dx, you know, from 0 to x,
- is equal to capital F of x.
- So it's kind of borrowing that notation, because this
- function of s is kind of an integral of y of t.
- The Laplace Transform, to some degree, is like a special type
- of integral where you have a little exponential function in
- there to mess around with things a little bit.
- Anyway, I just wanted you to get used to this notation.
- When you see capital Y of s, that's the same thing as a
- Laplace Transform of y of t.
- And you might also see it this way.
- The Laplace Transform of f of t is equal to capital F of s.
- And the clue that tells you that this isn't just a normal
- antiderivative, is the fact that they're using that s as
- the independent variable.
- Because in general, s represents the frequency
- domain, and if people were to use s with just a general
- antiderivative, people would get confused,
- et cetera, et cetera.
- Anyway, I'm trying to think whether I have time to teach
- you more fascinating concepts of Laplace Transform.
- Well, sure, I think we do.
- So my next question for you-- and now we'll teach you a
- couple more properties, and this'll be helpful in taking
- Laplace Transforms. What is the Laplace Transform of e to
- the at times f of t?
- Fascinating.
- Well, let's just should go back to our definition of the
- Laplace Transform.
- It is the integral from 0 to infinity of e to the minus st
- times whatever we have between the curly brackets.
- So, with the curly brackets we have e to the at f of t dt.
- And now we can add these exponents.
- We have a similar base, so this is equal to what?
- This is equal to the integral from 0 to infinity.
- And let's see, I want to write it as, I could write it minus
- s plus a, but I'm going to write it as minus s minus a t.
- And you could expand this out.
- It becomes minus s plus a, which is exactly what we have
- here, times f of t dt.
- Now let me show you something.
- if I were to just take the Laplace Transform of f of t,
- that is equal to some function of s.
- Whatever we essentially have right here for s, it becomes
- some function of that.
- So this is interesting.
- This is some function of s.
- Here, all we did to go from-- well actually
- let me rewrite this.
- The Laplace, which is equal to 0 to infinity e to the minus
- st f of t dt.
- The Laplace Transform of just f of t is equal to this, which
- is some function of s.
- Well, the Laplace Transform of e to the at, times f of t, it
- equals this.
- And what's the difference between this and this?
- What's the difference between the two?
- Well, it's not much.
- Here, wherever I have an s, I have an s minus a here.
- So if this is a function of s, what's this going to be?
- It's going to be that same function.
- Whatever the Laplace Transform of f was, it's going to be
- that same function, but instead of s, it's going to be
- a function of s minus a.
- And once again, how did I get that?
- Well I said the Laplace Transform of f is a function
- of s, and it's equal to this.
- Well if I just replace an s with an s minus a, I get this,
- which is a function of s minus a.
- Which was the Laplace Transform of e to the
- at times f of t.
- Maybe that's a little confusing.
- Let me show you an example.
- Let's just take the Laplace Transform of cosine of 2t.
- We've shown is equal to-- well I'll write the notation-- it's
- equal to some function of s.
- And that function of s is s over s squared plus 4.
- We've shown that already.
- And so the Laplace Transform of e to the, I don't know, 3t
- times cosine of 2t, is going to be equal to the same
- function, but instead of s, it's going to be a
- function of s minus a.
- So s minus 3, which is equal to s minus 3 over s minus 3
- squared plus 4.
- Notice, when you just multiply something by this, either the
- 3t and then or either the at, you take the Laplace Transform
- of it, you just-- it's the same thing as the Laplace
- Transform of this function, but everywhere where you had
- an s, you replace it with an s minus this a.
- Anyway, I hope I didn't confuse you too much
- with that last part.
- I think my power adaptor actually just went on.
- I hope the video keeps recording.
- I'll see you in the next one.

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