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- I'll now introduce you to the concept
- of the Laplace Transform.
- And this is truly one of the most useful concepts that
- you'll learn, not just in differential equations, but
- really in mathematics.
- And especially if you're going to go into engineering, you'll
- find that the Laplace Transform, besides helping you
- solve differential equations, also helps you transform
- functions or waveforms from the time domain to the
- frequency domain, and study and understand a
- whole set of phenomena.
- But I won't get into all of that yet.
- Now I'll just teach you what it is.
- Laplace Transform.
- I'll teach you what it is, make you comfortable with the
- mathematics of it and then in a couple of videos from now,
- I'll actually show you how it is useful to use it to solve
- differential equations.
- We'll actually solve some of the differential equations we
- did before, using the previous methods.
- But we'll keep doing it, and we'll solve more and more
- difficult problems.
- So what is the Laplace Transform?
- Well, the Laplace Transform, the notation is the L like
- Laverne from Laverne and Shirley.
- That might be before many of your times, but
- I grew up on that.
- Actually, I think it was even reruns when I was a kid.
- So Laplace Transform of some function.
- And here, the convention, instead of saying f of x,
- people say f of t.
- And the reason is because in a lot of the differential
- equations or a lot of engineering you actually are
- converting from a function of time to
- a function of frequency.
- And don't worry about that right now.
- If it confuses you.
- But the Laplace Transform of a function of t.
- It transforms that function into some other function of s.
- and And does it do that?
- Well actually, let me just do some mathematical notation
- that probably won't mean much to you.
- So what does it transform?
- Well, the way I think of it is it's kind of
- a function of functions.
- A function will take you from one set of-- well, in what
- we've been dealing with-- one set of numbers to another set
- of numbers.
- A transform will take you from one set of functions to
- another set of functions.
- So let me just define this.
- The Laplace Transform for our purposes is defined as the
- improper integral.
- I know I haven't actually done improper integrals just yet,
- but I'll explain them in a few seconds.
- The improper integral from 0 to infinity of e to the minus
- st times f of t-- so whatever's between the Laplace
- Transform brackets-- dt.
- Now that might seem very daunting to you and very
- confusing, but I'll now do a couple of examples.
- So what is the Laplace Transform?
- Well let's say that f of t is equal to 1.
- So what is the Laplace Transform of 1?
- So if f of t is equal to 1-- it's just a constant function
- of time-- well actually, let me just substitute exactly the
- way I wrote it here.
- So that's the improper integral from 0 to infinity of
- e to the minus st times 1 here.
- I don't have to rewrite it here, but there's a times 1dt.
- And I know that infinity is probably bugging you right
- now, but we'll deal with that shortly.
- Actually, let's deal with that right now.
- This is the same thing as the limit.
- And let's say as A approaches infinity of the integral from
- 0 to Ae to the minus st. dt.
- Just so you feel a little bit more comfortable with it, you
- might have guessed that this is the same thing.
- Because obviously you can't evaluate infinity, but you
- could take the limit as something approaches infinity.
- So anyway, let's take the anti-derivative and evaluate
- this improper definite integral, or
- this improper integral.
- So what's anti-derivative of e to the minus st
- with respect to dt?
- Well it's equal to minus 1/s e to the minus st, right?
- If you don't believe me, take the derivative of this.
- You'd take minus s times that.
- That would all cancel out, and you'd just be left with e to
- the minus st. Fair enough.
- Let me delete this here, this equal sign.
- Because I could actually use some of that real estate.
- We are going to take the limit as A approaches infinity.
- You don't always have to do this, but this is the first
- time we're dealing with improper intergrals.
- So I figured I might as well remind you that
- we're taking a limit.
- Now we took the anti-derivative.
- Now we have to evaluate it at A minus the anti-derivative
- evaluate it at 0,
- and then take the limit of whatever that ends up being as
- A approaches infinity.
- So this is equal to the limit as A approaches infinity.
- OK.
- If we substitute A in here first, we get minus 1/s.
- Remember we're, dealing with t.
- We took the integral with respect to t.
- e to the minus sA, right?
- That's what happens when I put A in here.
- Minus -
- Now what happens when I put t equals 0 in here?
- So when t equals 0, it becomes e to the minus s times 0.
- This whole thing becomes 1.
- And I'm just left with minus 1/s.
- Fair enough.
- And then let me scroll down a little bit.
- I wrote a little bit bigger than I wanted
- to, but that's OK.
- So this is going to be the limit as A approaches infinity
- of minus 1/s e to the minus sA minus minus 1/s.
- So plus 1/s.
- So what's the limit as A approaches infinity?
- Well what's this term going to do?
- As A approaches infinity, if we assume that s is greater
- than 0-- and we'll make that assumption for now.
- Actually, let me write that down explicitly.
- Let's assume that s is greater than 0.
- So if we assume that s is greater than 0, then as A
- approaches infinity, what's going to happen?
- Well this term is going to go to 0, right? e to the minus--
- a googol is a very, very small number.
- And an e to the minus googolplex is an even smaller number.
- So then this e to the minus infinity approaches 0, so this
- term approaches 0.
- This term isn't affected because it has no A in it, so
- we're just left with 1/s.
- So there you go.
- This is a significant moment in your life.
- You have just been exposed to your first Laplace Transform.
- I'll show you in a few videos, there are whole tables of
- Laplace Transforms, and eventually we'll
- prove all of them.
- But for now, we'll just work through some of
- the more basic ones.
- But this can be our first entry in our
- Laplace Transform table.
- The Laplace Transform of f of t is equal to
- 1 is equal to 1/s.
- Notice we went from a function of t-- although obviously this
- one wasn't really dependent on t-- to a function of s.
- I have about 3 minutes left, but I don't think that's
- enough time to do another Laplace Transform.
- So I will save that for the next video.
- See you soon.