Laplace transform to solve a differential equation
Laplace Transform to solve an equation Using the Laplace Transform to solve an equation we already knew how to solve.
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- I've been doing a ton of videos on the mechanics of
- taking the Laplace Transform, but you've been sitting
- through them always wondering, what am I learning this for?
- And now I'll show you, at least in the context of
- differential equations.
- And I've gotten a bunch of letters
- on the Laplace Transform.
- What does it really mean?
- And and all that.
- And those are excellent questions and you should
- strive for that.
- It's hard to really have an intuition of the Laplace
- Transform in the differential equations context, other than
- it being a very useful tool that converts differential or
- integral problems into algebra problems. But I'll give you a
- hint, and if you want a path to learn it in, you should
- learn about Fourier series and Fourier Transforms, which are
- very similar to Laplace Transforms. And that'll
- actually build up the intuition on what the
- frequency domain is all about.
- Well anyway, let's actually use the Laplace Transform to
- solve a differential equation.
- And this is one we've seen before.
- So let me see.
- So let's say the differential equation is y prime prime,
- plus 5, times the first derivative, plus
- 6y, is equal to 0.
- And you know how to solve this one, but I just want to show
- you, with a fairly straightforward differential
- equation, that you could solve it with the Laplace Transform.
- And actually, you end up having a
- characteristic equation.
- And the initial conditions are y of 0 is equal to 2, and y
- prime of 0 is equal to 3.
- Now, to use the Laplace Transform here, we essentially
- just take the Laplace Transform of both sides of
- this equation.
- Let me use a more vibrant color.
- So we get the Laplace Transform of y the second
- derivative, plus-- well we could say the Laplace
- Transform of 5 times y prime, but that's the same thing as 5
- times the Laplace Transform-- y prime.
- y prime plus 6 times the Laplace Transform of y.
- And let me ask you a question.
- What's the Laplace Transform of 0?
- Let me do that.
- So the Laplace Transform of 0 would be be the integral from
- 0 to infinity, of 0 times e to the minus stdt.
- So this is a 0 in here.
- So this is equal to 0.
- So the Laplace Transform of 0 is 0.
- And that's good, because I didn't have space to do
- another curly L.
- So what are the Laplace Transforms of these things?
- Well this is where we break out one of the useful
- properties that we learned.
- Let me write it over here.
- I think that's going to need as much
- real estate as possible.
- Let me erase this.
- So we learned that the Laplace Transform-- I'll do it here.
- Actually, I'll do it down here.
- The Laplace Transform of f prime, or we could even say y
- prime, is equal to s times the Laplace Transform of
- y, minus y of 0.
- We proved that to you.
- And this is extremely important to know.
- So let's see if we can apply that.
- So the Laplace Transform of y prime prime, if we apply that,
- that's equal to s times the Laplace Transform of-- well if
- we go from y prime to y, you're just taking the
- anti-derivative, so if you're taking the anti-derivative of
- y, of the second derivative, we just end up with the first
- derivative-- minus the first derivative at 0.
- Notice, we're already using our initial conditions.
- I won't substitute it just yet.
- And then we end up with plus 5, times-- I'll write it every
- time-- so plus 5 times the Laplace Transform of y prime,
- plus 6 times the Laplace Transform of y.
- All of that is equal to 0.
- So just to be clear, all I did is I expanded this into this
- using this.
- So how can we rewrite the Laplace Transform of y prime?
- Well, we could use this once again, so let's do that.
- So this over here-- I'll do it in magenta-- this is equal to
- s times what?
- s times the Laplace Transform of y prime.
- Well that's s times the Laplace Transform of y, minus
- y of 0, right?
- I took this part and replaced it with what I have in
- parentheses.
- So minus y prime of 0-- and now I'll switch colors-- plus
- 5 times-- once again the Laplace Transform of y prime.
- Well we can use this again.
- So 5 times s times Laplace Transform of y, minus y of 0,
- plus 6 times the Laplace Transform-- oh I ran out of
- space, I'll do it in another line-- plus 6 times the
- Laplace Transform of y.
- All of that is equal to 0.
- I know this looks really confusing but we'll
- simplify right now.
- And we could get rid of this right here, because we've used
- it as much as we need to.
- So now we just simplify.
- And notice, using the Laplace Transform, we didn't have to
- guess at a general solution or anything like that.
- Even when we did a characteristic equation, we
- guessed what the original general solution was.
- Now we're just taking Laplace Transforms, and let's see
- where this gets us.
- And actually I just want to make clear, because I know
- it's very confusing, so I rewrote this part as this.
- And I rewrote this thing as this.
- And everything else is the same.
- But now let's simplify the math.
- So we get s squared, times the Laplace Transform of y-- I'm
- going to write smaller, I've learned my lesson-- minus s
- times y of 0.
- Let's substitute y of 0 here. y of 0 is 2, so s times y of 0
- is 2 times s, so 2s, distribute that s, minus y
- prime of 0.
- Y prime of 0 is 3.
- So minus 3, plus-- so we have 5 times s times the Laplace
- Transform of y, so plus 5s times the Laplace Transform of
- y, minus 5 times y of 0. y of 0 is 2, so minus 10.
- Minus 10, right?
- 5 times-- this is 2 right here-- so 5 times 2, plus 6
- times the Laplace Transform of y.
- All of that is equal to 0.
- Now, let's group our Laplace Transform of y terms and our
- constant terms, and we should be hopefully
- getting some place.
- So let's see, my Laplace Transform of y terms, I have
- this one, I have this one, and I have that one.
- So what am I left with?
- Well let me factor out the Laplace Transform of y part.
- So I get the Laplace Transform of y-- and that's good because
- it's a pain to keep writing it over and over-- times s
- squared plus 5s plus 6.
- So those are all my Laplace Transform terms. And then I
- have my constant terms. So let's see, I have 1s, so minus
- 2s, minus 3, minus 10, is equal to 0.
- And what can we do here?
- Well, this is interesting, first of all.
- Notice that the coefficients on the Laplace Transform of y
- terms, that those are that characteristic equation that
- we dealt with so much, and that is hopefully, to some
- degree, second nature to you.
- So that's a little bit of a clue, and if you want some
- very tenuous connections, well that makes a lot of sense.
- Because the characteristic equation to get that, we
- substituted e to the rt, and the Laplace Transform involves
- very similar function.
- But anyway, let's go back to the problem.
- So how do we solve this?
- And actually, let me just give you the big picture here,
- because this is a good point.
- What I'm going to do is I'm going to solve this.
- I'm going to say the Laplace Transform of
- y is equal to something.
- And then I'm going to say, boy, what functions the
- Laplace Transform is at something?
- And then I'll have the solution.
- If that confuses you, just wait and hopefully it'll make
- some sense.
- From here until that point it's just some
- fairly hairy algebra.
- So let's scroll down a little bit, just so we have some
- breathing room.
- And so I get the Laplace Transform of y, times s
- squared, plus 5s, plus 6, is equal to-- let's add these
- terms to both sides of this equation-- is equal to 2s plus
- 3 plus 10-- oh, that's silly-- plus 13.
- This is minus 13 here.
- A phone call.
- Who's calling?
- I think it's some kind of marketing phone call.
- Anyway, 2s plus 13, and now what can I do?
- Well.
- Let's divide both sides by this s squared plus 5s plus 6.
- So I get the Laplace Transform of y is equal to 2s plus 13,
- over s squared plus 5s plus 6.
- Now we're almost done.
- Everything here is just a little bit of algebra.
- So now we're almost done.
- We haven't solved for y yet, but we know that the Laplace
- Transform of y is equal to this.
- Now, if we just had this in our table of our Laplace
- Transforms, we would immediately know what y was,
- but I don't see something, or I don't remember anything we
- did in our table that looks like this expression of s.
- I'm essentially out of time, so the next video we're going
- to figure out what functions Laplace Transform is this.
- And it actually turns out it's a sum of things we already
- know, and we just have to manipulate this a little bit
- algebraically.
- See you in the next video.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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