Intro to differential equations
What is a differential equation What a differential equation is and some terminology.
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- Welcome to this first video, and actually the first video
- in the playlist on differential equations.
- I know I touched on this before when we did harmonic
- motion, and I think I might have touched
- on it in other subjects.
- But now, because of your request, we'll do a whole
- playlist on this.
- And that's a fairly useful thing, because differential
- equations is something that shows up in a whole set of
- different fields.
- I've been requested by someone who's starting an economics
- PhD program to do this; I've been requested by some people
- who are going into physics, some people who are going into
- engineering.
- So it's a widely applicable area of study.
- So let's just get started, before I keep going off on
- useless stuff.
- So the differential equations.
- So the first question is: what is a differential equation?
- You know what an equation is.
- What is a differential equation?
- Well, a differential equation is an equation that involves
- an unknown function and its derivatives.
- So what do I mean by that?
- Well, let's say that I said that y prime plus y is equal
- to x plus 3.
- Here, the unknown function is y.
- We could have written it as y of x, or we could have written
- this as dy dx, the derivative of y with respect to x plus
- this unknown function y is equal to x plus 3.
- We also could have written f prime of x plus f of x is
- equal to x plus 3.
- All of these would have been valid ways of writing this
- exact same differential equation.
- And what's interesting here, and how this is a departure
- from what we've learned before about just regular equations
- is that-- let me write down a regular equation just to
- remind you what they look like.
- So a regular equation, if we had one variable, would look
- something like this.
- I don't know, x squared plus the cosine of x is equal to
- the square root of x.
- I just made that up.
- Here, the solution is a number, or sometimes it's a
- set of numbers.
- Sometimes there's more than one, right?
- If you have a polynomial, you could have more than one
- values of x that satisfy this equation.
- Here, for a differential equation, the
- solution is a function.
- Our goal is to figure out what function of x, and here I
- wrote f of x explicitly, but what function of x explicitly
- satisfies this relationship or this equation.
- So let me show you what I mean by that.
- And I have my differential equations book from college,
- so I'm going to use that as we go.
- So let's say that-- I'm just writing now.
- See, they have this as a differential equation.
- And I'm not going to show you necessarily how to solve them
- just yet, because we have to learn some tricks first. But I
- think a good place to start is just so you understand what a
- differential equation is, so you don't get confused with
- the traditional equation.
- So, they have this differential [? derivative. ?]
- y prime prime.
- So the second derivative of y with respect to x, plus 2
- times the first derivative of y with respect to x, minus 3 y
- is equal to 0.
- And they give us the solutions here, and what they want us to
- do is show that these are solutions.
- And I think this is a good place to just at least
- understand what a differential equation is, and what its
- solution means.
- So they say y1 of x is equal to e to the minus 3x.
- So they claim that this is a solution of this
- differential equation.
- So let me show to you that this is.
- Well, if this is soon. y1, what's y-- well, let
- me just write y1.
- What's y1 prime?
- What's the derivative of this?
- Well, just do the chain rule.
- The derivative of the whole function, with respect to this
- part of it, is just e to the minus 3x.
- And then you take the derivative of the inside.
- So that's just the derivative of the outside, e
- to the minus 3x.
- And the derivative of the inside is minus 3.
- And the second derivative of y1 is equal to-- we'll just
- take the derivative of this, and that's just equal to plus
- 9-- minus 3 times minus 3-- e to the minus 3x.
- Now, let's verify that if we substitute y1 and its
- derivatives back into this differential equation, that it
- holds true.
- So y prime prime, that's this.
- So we get nine e to the minus 3x, plus 2y prime.
- Plus 2 times y prime.
- Well, this is y prime.
- So 2 times minus 3 e to the minus 3x plus-- oh sorry,
- minus-- 3 times y.
- Well, y is this.
- So minus 3 times e to the minus 3x.
- Well, what does that equal?
- We get 9 e to the minus 3x, minus 6 e to the minus 3x,
- minus 3 e to the minus 3x.
- Well, what does that equal?
- We have 9 of something minus 6 of
- something minus 3 of something.
- So that just equals 0.
- It doesn't matter of 0 whatever.
- So that equals 0.
- So we verified that for this function, for y1 is equal to e
- to the minus 3x, it satisfies this differential equation.
- Now there's something interesting here, and you've
- kind of touched on this with regular equations, is that
- this might not be the only solution.
- In fact we'll learn, in maybe a video or two, that often the
- solution is not just a function.
- It could be a class of functions where usually
- they're all kind of the same function, but you have a
- difference of constants.
- But I'll show you that in a second.
- But here, they actually show us that there's another
- solution, that this will actually work with, we could
- try the equation y2 of x is equal to, well, just
- simple e to the x.
- And we could verify that, right?
- What's the first and second derivatives of e to the x?
- Well, they're just e to the x.
- So the second derivative of y2 is just e to the x plus 2
- times the first derivative is what?
- Well the first derivative of e to the x is still e to the x,
- 2 e to the x, minus 3 times a function.
- Minus 3e to the x.
- Well, 1 plus 2 minus 3, well that equals 0 again.
- So this was also a solution to this differential equation.
- Now before we go on, in the next one I'll show you some
- fairly straightforward differential
- equations to solve.
- I think it's a good time now, now that you hopefully have a
- grasp of what a differential equation is, and what its
- solution is.
- And its solution isn't a number, its solution is a
- function, or a set of functions,
- or a class of functions.
- It's a good time to just go over a little bit of
- terminology.
- So there's two big classifications.
- Well actually, there's a first big one, ordinary and partial
- differential equations.
- I think you might have already guessed what that means.
- An ordinary differential equation is what I wrote down.
- It's one variable with respect to another variable, or one
- function with respect you to, say, x and its derivatives.
- Partial differential equations we'll get into later.
- That's more complicated.
- That's when a function can be a function of
- more than one variable.
- And you can have the derivative with respect to x,
- and y, and z.
- We won't worry about that right now.
- If your functions and their derivatives are a function of
- only one variable, then we're dealing with an ordinary
- differential equation.
- That's what this playlist will deal with, ordinary
- differential equations.
- Now within ordinary differential equations,
- there's two ways of classifying, and
- they kind of overlap.
- You have your order, so what is the order of my
- differential equation?
- And then you have this notion of whether it is linear or
- non-linear.
- And I think the best way to figure this out is just to
- write down examples.
- So let me write down one.
- And I'm getting this from my college
- differential equations book.
- x squared times the second derivative of y with respect
- to x, plus x times the first derivative of y with respect
- to x, plus 2y is equal to sine of x.
- So the first question here is: what is the order?
- All the order is is the highest derivative that exists
- in your equation.
- The highest derivative of the function
- under question, right?
- The solution of this is going to be a y of x, that satisfies
- this equation.
- And the order is the highest derivative of that function.
- Well, the highest derivative here is the second derivative.
- So this has order 2.
- Or as you could call this, a second order ordinary
- differential equation.
- Now the second thing we have to figure out: is this linear
- or is this a non-linear differential equation?
- So a differential equation is linear if all of the functions
- and its derivatives are essentially, well for lack of
- a better word, linear.
- What do I mean by that?
- I mean you don't have a y squared, or you don't have a
- dy over dx squared, or you don't have a y times the
- second derivative of y.
- So this example I just wrote here, this is a second order
- linear equation, because you have the second derivative,
- the first derivative, and y, but they're not multiplied by
- the function or the derivatives.
- Now if this equation were-- if I rewrote it as x squared d,
- the second derivative of y with respect to x squared, is
- equal to sine of x, and let's say I were to square this.
- Now, all of the sudden, I have a non-linear
- differential equation.
- This is non-linear.
- This is linear.
- Because I squared, I multiplied the second
- derivative of y with respect-- I multiplied it times itself.
- Another example of a non-linear equation is if I
- wrote y times the second derivative of y with respect
- to x is equal to sine of x.
- This is also non-linear, because I multiplied the
- function times its second derivative.
- Notice here, I did multiply stuff times the second
- derivative, but it was the independent variable x that I
- multiplied.
- But anyway, I've run out of time, and hopefully that gives
- you a good at least survey of what a
- differential equation is.
- In the next video, we'll start actually solving them.
- See you soon
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