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 plus 3.
We also could have written f prime of x plus f of x is
equal to 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