Introduction to differential equations

What a differential equation is and some terminology.

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