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Welcome to this first video, and actually the first video

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in the playlist on differential equations.

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I know I touched on this before when we did harmonic

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motion, and I think I might have touched

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on it in other subjects.

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But now, because of your request, we'll do a whole

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playlist on this.

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And that's a fairly useful thing, because differential

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equations is something that shows up in a whole set of

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different fields.

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I've been requested by someone who's starting an economics

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PhD program to do this; I've been requested by some people

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who are going into physics, some people who are going into

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engineering.

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So it's a widely applicable area of study.

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So let's just get started, before I keep going off on

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useless stuff.

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So the differential equations.

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So the first question is: what is a differential equation?

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You know what an equation is.

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What is a differential equation?

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Well, a differential equation is an equation that involves

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an unknown function and its derivatives.

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So what do I mean by that?

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Well, let's say that I said that y prime plus y is equal

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to x plus 3.

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Here, the unknown function is y.

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We could have written it as y of x, or we could have written

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this as dy dx, the derivative of y with respect to x plus

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this unknown function y is equal to plus 3.

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We also could have written f prime of x plus f of x is

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equal to plus 3.

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All of these would have been valid ways of writing this

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exact same differential equation.

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And what's interesting here, and how this is a departure

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from what we've learned before about just regular equations

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is that-- let me write down a regular equation just to

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remind you what they look like.

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So a regular equation, if we had one variable, would look

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something like this.

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I don't know, x squared plus the cosine of x is equal to

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the square root of x.

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I just made that up.

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Here, the solution is a number, or sometimes it's a

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set of numbers.

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Sometimes there's more than one, right?

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If you have a polynomial, you could have more than one

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values of x that satisfy this equation.

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Here, for a differential equation, the

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solution is a function.

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Our goal is to figure out what function of x, and here I

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wrote f of x explicitly, but what function of x explicitly

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satisfies this relationship or this equation.

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So let me show you what I mean by that.

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And I have my differential equations book from college,

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so I'm going to use that as we go.

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So let's say that-- I'm just writing now.

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See, they have this as a differential equation.

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And I'm not going to show you necessarily how to solve them

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just yet, because we have to learn some tricks first. But I

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think a good place to start is just so you understand what a

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differential equation is, so you don't get confused with

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the traditional equation.

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So, they have this differential [? derivative. ?]

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y prime prime.

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So the second derivative of y with respect to x, plus 2

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times the first derivative of y with respect to x, minus 3 y

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is equal to 0.

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And they give us the solutions here, and what they want us to

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do is show that these are solutions.

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And I think this is a good place to just at least

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understand what a differential equation is, and what its

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solution means.

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So they say y1 of x is equal to e to the minus 3x.

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So they claim that this is a solution of this

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differential equation.

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So let me show to you that this is.

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Well, if this is soon. y1, what's y-- well, let

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me just write y1.

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What's y1 prime?

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What's the derivative of this?

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Well, just do the chain rule.

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The derivative of the whole function, with respect to this

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part of it, is just e to the minus 3x.

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And then you take the derivative of the inside.

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So that's just the derivative of the outside, e

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to the minus 3x.

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And the derivative of the inside is minus 3.

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And the second derivative of y1 is equal to-- we'll just

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take the derivative of this, and that's just equal to plus

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9-- minus 3 times minus 3-- e to the minus 3x.

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Now, let's verify that if we substitute y1 and its

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derivatives back into this differential equation, that it

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holds true.

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So y prime prime, that's this.

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So we get nine e to the minus 3x, plus 2y prime.

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Plus 2 times y prime.

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Well, this is y prime.

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So 2 times minus 3 e to the minus 3x plus-- oh sorry,

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minus-- 3 times y.

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Well, y is this.

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So minus 3 times e to the minus 3x.

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Well, what does that equal?

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We get 9 e to the minus 3x, minus 6 e to the minus 3x,

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minus 3 e to the minus 3x.

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Well, what does that equal?

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We have 9 of something minus 6 of

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something minus 3 of something.

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So that just equals 0.

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It doesn't matter of 0 whatever.

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So that equals 0.

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So we verified that for this function, for y1 is equal to e

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to the minus 3x, it satisfies this differential equation.

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Now there's something interesting here, and you've

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kind of touched on this with regular equations, is that

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this might not be the only solution.

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In fact we'll learn, in maybe a video or two, that often the

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solution is not just a function.

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It could be a class of functions where usually

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they're all kind of the same function, but you have a

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difference of constants.

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But I'll show you that in a second.

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But here, they actually show us that there's another

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solution, that this will actually work with, we could

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try the equation y2 of x is equal to, well, just

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simple e to the x.

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And we could verify that, right?

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What's the first and second derivatives of e to the x?

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Well, they're just e to the x.

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So the second derivative of y2 is just e to the x plus 2

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times the first derivative is what?

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Well the first derivative of e to the x is still e to the x,

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2 e to the x, minus 3 times a function.

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Minus 3e to the x.

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Well, 1 plus 2 minus 3, well that equals 0 again.

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So this was also a solution to this differential equation.

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Now before we go on, in the next one I'll show you some

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fairly straightforward differential

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equations to solve.

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I think it's a good time now, now that you hopefully have a

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grasp of what a differential equation is, and what its

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solution is.

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And its solution isn't a number, its solution is a

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function, or a set of functions,

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or a class of functions.

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It's a good time to just go over a little bit of

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terminology.

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So there's two big classifications.

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Well actually, there's a first big one, ordinary and partial

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differential equations.

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I think you might have already guessed what that means.

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An ordinary differential equation is what I wrote down.

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It's one variable with respect to another variable, or one

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function with respect you to, say, x and its derivatives.

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Partial differential equations we'll get into later.

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That's more complicated.

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That's when a function can be a function of

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more than one variable.

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And you can have the derivative with respect to x,

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and y, and z.

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We won't worry about that right now.

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If your functions and their derivatives are a function of

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only one variable, then we're dealing with an ordinary

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differential equation.

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That's what this playlist will deal with, ordinary

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differential equations.

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Now within ordinary differential equations,

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there's two ways of classifying, and

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they kind of overlap.

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You have your order, so what is the order of my

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differential equation?

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And then you have this notion of whether it is linear or

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non-linear.

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And I think the best way to figure this out is just to

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write down examples.

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So let me write down one.

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And I'm getting this from my college

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differential equations book.

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x squared times the second derivative of y with respect

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to x, plus x times the first derivative of y with respect

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to x, plus 2y is equal to sine of x.

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So the first question here is: what is the order?

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All the order is is the highest derivative that exists

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in your equation.

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The highest derivative of the function

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under question, right?

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The solution of this is going to be a y of x, that satisfies

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this equation.

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And the order is the highest derivative of that function.

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Well, the highest derivative here is the second derivative.

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So this has order 2.

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Or as you could call this, a second order ordinary

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differential equation.

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Now the second thing we have to figure out: is this linear

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or is this a non-linear differential equation?

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So a differential equation is linear if all of the functions

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and its derivatives are essentially, well for lack of

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a better word, linear.

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What do I mean by that?

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I mean you don't have a y squared, or you don't have a

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dy over dx squared, or you don't have a y times the

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second derivative of y.

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So this example I just wrote here, this is a second order

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linear equation, because you have the second derivative,

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the first derivative, and y, but they're not multiplied by

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the function or the derivatives.

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Now if this equation were-- if I rewrote it as x squared d,

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the second derivative of y with respect to x squared, is

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equal to sine of x, and let's say I were to square this.

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Now, all of the sudden, I have a non-linear

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differential equation.

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This is non-linear.

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This is linear.

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Because I squared, I multiplied the second

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derivative of y with respect-- I multiplied it times itself.

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Another example of a non-linear equation is if I

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wrote y times the second derivative of y with respect

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to x is equal to sine of x.

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This is also non-linear, because I multiplied the

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function times its second derivative.

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Notice here, I did multiply stuff times the second

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derivative, but it was the independent variable x that I

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multiplied.

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But anyway, I've run out of time, and hopefully that gives

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you a good at least survey of what a

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differential equation is.

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In the next video, we'll start actually solving them.

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See

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