Exact equations example 2

WEBVTT
Kind: captions
Language: en

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Let's do some more examples
with exact

00:00:02.770 --> 00:00:03.670
differential equations.

00:00:03.670 --> 00:00:08.300
And I'm getting these problems
from page 80 of my old college

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differential equations books.

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This is the fifth edition of
Elementary Differential

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Equations by William Boyce
and Richard DiPrima.

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I want to make sure they get
credit, that I'm not making up

00:00:17.800 --> 00:00:19.980
these problems. I'm getting
it from their book.

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Anyway, so I'm just going to
give a bunch of equations.

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We have to figure out if they're
exact, and if they are

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exact, we'll use what we know
about exact differential

00:00:26.850 --> 00:00:29.070
equations to figure out
their solutions.

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So the first one they have is,
2x plus 3, plus 2y minus 2,

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times y prime is equal to 0.

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So this is our M of x and y--
although, this is only a

00:00:47.440 --> 00:00:50.360
function of x-- and then
this is our N, right?

00:00:50.360 --> 00:00:53.130
You could say that's
M, or that's N.

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You could also say that, if this
is exact-- well, first

00:00:55.560 --> 00:00:55.873
let's [? test ?]

00:00:55.873 --> 00:00:58.120
this exact, before we start
talking about psi.

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So what's the partial of this,
with respect to y?

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The partial of M, with
respect to y.

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Well, there's no y
here, so it's 0.

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The rate of change that
this changes with

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respect to y is 0.

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And what's the rate of
change this changes,

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with respect to x?

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The partial of N, with respect
to x is equal to-- well,

00:01:18.140 --> 00:01:19.490
there's no x here, right?

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So these are just constants from
an x point of view, so

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this is all going to be 0.

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But we do see that
they're both 0.

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So M sub y, or the partial with
respect to y, is equal to

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the partial with respect to x.

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So this is exact.

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And actually, we don't have to
use exact equations here.

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We'll do it, just so that
we get used to it.

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But if you look here, you
actually could have figured

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out that this is actually
a separable equation.

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But anyway, this is exact.

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So knowing that it's exact, it
tells us that there's some

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function psi, where psi is
a function of x and y.

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Where psi sub x is equal to this
function, is equal to 2x

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plus 3, and psi-- I shouldn't
say sub x.

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I say the partial of psi,
with respect to x.

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And the partial of psi, with
respect to y, is equal to

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this, 2y minus 2.

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And if we can find our psi, we
know that this is just the

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derivative of psi.

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Because we know that the
derivative, with respect to x

00:02:20.290 --> 00:02:24.510
of psi, is equal to the partial
of psi, with respect

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to x, plus the partial
of psi, with respect

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to y, times y prime.

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So this is this just the
same form as that.

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So if we can figure out y,
then we can rewrite this

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equation as dx, the derivative
of psi, with respect to x, is

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equal to 0.

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Let me switch colors, or it's
going to get monotonous.

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This right here, if we can find
a psi, where the partial

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with respect to x, is this, the
partial with respect to y,

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is this, then this can
be rewritten as this.

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And how do we know that?

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Because the derivative of psi,
with respect to x, using the

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partial derivative chain
rules, is this.

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This partial with respect
to x, that's this.

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This partial with respect to
y, is this, times y prime.

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So this is the whole point
of exact equations.

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But anyway, so let's figure
out what our psi is.

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Actually, before we figure out,
if the derivative of psi,

00:03:21.170 --> 00:03:23.410
with respect to x, is 0, then
if you integrate both sides,

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you just-- the solution
of this equation is

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psi is equal to c.

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So using this information, if we
can solve for psi, then we

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know that the solution of this
differential equation is psi

00:03:39.630 --> 00:03:41.140
is equal to c.

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And then if we have some initial
conditions, we could

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solve for c.

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So let's solve for psi.

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So let's integrate both sides
of this equation,

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with respect to x.

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And then we get psi is equal
to x squared plus 3x, plus

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some function of y.

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Let's call it h of y.

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And remember, normally when you
take an antiderivative,

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you have just a plus
c here, right?

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But you can kind of say we
took an anti-partial

00:04:09.210 --> 00:04:10.560
derivative.

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So when you took a partial
derivative, with respect to x,

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not only do you lose constants--
that's why we have

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a plus c, normally-- but you
also lose anything that's a

00:04:17.870 --> 00:04:20.570
function of just y, and not x.

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So for example, take the partial
derivative of this

00:04:23.110 --> 00:04:25.130
with respect to x, you're going
to get this, right?

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Because the partial derivative
of a function, purely of y,

00:04:27.430 --> 00:04:29.090
with respect to x,
is going to be 0.

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So it will disappear.

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So anyway, we take the
antiderivative of

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this, we get this.

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Now, we use this information.

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Well, we use this information.

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We take the partial of this
expression, and we say, well,

00:04:40.240 --> 00:04:44.000
the partial of this expression,
with respect to y,

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has to equal this, and then we
can solve for h of y, then

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we'll be done.

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So let's do that.

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So the partial of psi, with
respect to y, is equal to--

00:04:52.920 --> 00:04:56.980
well, that's going
to be 0, 0, 0.

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This part is a function of x.

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If you take the partial with
respect to y, it's 0, because

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these are constants, from
a y point of view.

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So you're left with
h prime of y.

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So we know that h prime of y,
which is the partial of psi,

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with respect to y,
is equal to this.

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So h prime of y is equal
to 2y minus 2.

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And then if we wanted to figure
out what h of y is, we

00:05:23.580 --> 00:05:26.180
get h of y-- just integrate both
sides, with respect to

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y-- is equal to y squared
plus-- sorry-- y

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squared minus 2y.

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Now, you could have a plus c
there, but if you watched the

00:05:35.340 --> 00:05:37.410
previous example, you'll see
that that c kind of merges

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with the other c, so you
don't have to worry

00:05:38.740 --> 00:05:39.750
about it right now.

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So what is our psi function,
as we know it now, not

00:05:43.540 --> 00:05:45.640
worrying about the plus c?

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It is psi of x and y is equal
to x squared plus 3x, plus h

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of y-- which we figured
out is this-- plus y

00:05:58.810 --> 00:06:02.100
squared, minus 2y.

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And we know a solution of our
original differential equation

00:06:05.450 --> 00:06:10.400
is psi is equal to c.

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So the solution of our
differential equation is this

00:06:13.280 --> 00:06:14.710
is equal to c.

00:06:14.710 --> 00:06:19.230
x squared plus 3x, plus
y squared, minus

00:06:19.230 --> 00:06:20.860
2y is equal to c.

00:06:20.860 --> 00:06:23.520
If you had some additional
conditions, you could test it.

00:06:23.520 --> 00:06:27.600
And I encourage you to test
this out on this original

00:06:27.600 --> 00:06:32.390
equation, or I encourage you to
take the derivative of psi,

00:06:32.390 --> 00:06:34.800
and prove to yourself that if
you took the derivative of

00:06:34.800 --> 00:06:39.750
psi, with respect to x, here,
implicitly, that you would get

00:06:39.750 --> 00:06:42.150
this differential equation.

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Anyway, let's do another one.

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Let's clear image.

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So the more examples you
see, the better.

00:06:49.190 --> 00:07:04.130
So let's see, this one says 2x
plus 4y, plus 2x minus 2y, y

00:07:04.130 --> 00:07:06.420
prime is equal to 0.

00:07:06.420 --> 00:07:10.340
So what's the partial of
this with respect to y?

00:07:10.340 --> 00:07:15.230
So M, the partial of M, with
respect to y-- this is 0-- so

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it's equal to 4.

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What's the partial of this, with
respect to x, just this

00:07:18.180 --> 00:07:19.970
part right here?

00:07:19.970 --> 00:07:23.460
The partial of N, with
respect to x, is 2.

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This is 0.

00:07:24.980 --> 00:07:28.090
So the partial of this, with
respect to y, is different

00:07:28.090 --> 00:07:29.920
than the partial of N,
with respect to x.

00:07:29.920 --> 00:07:33.500
So this is not exact.

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So we can't solve this using
our exact methodology.

00:07:37.360 --> 00:07:39.070
So that was a fairly
straightforward problem.

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Let's do another one.

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Let's see.

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I'm running out of time, so I
want to do one that's not too

00:07:44.710 --> 00:07:46.190
complicated.

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Let's see, 3x squared minus
2xy-- actually, let me do this

00:07:54.130 --> 00:07:54.810
in the next problem.

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I don't want to rush
these things.

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I will continue this
in the next video.

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See you soon.


Exact equations intuition 2 (proofy)

WEBVTT
Kind: captions
Language: en

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In the last video I introduced
you to the idea of the chain

00:00:04.470 --> 00:00:05.520
rule with partial derivatives.

00:00:05.520 --> 00:00:10.080
And we said, well, if I have a
function, psi, Greek letter,

00:00:10.080 --> 00:00:14.020
psi, it's a function
of x and y.

00:00:14.020 --> 00:00:16.770
And if I wanted to take the
partial of this, with respect

00:00:16.770 --> 00:00:19.360
to-- no, I want to take the
derivative, not the partial--

00:00:19.360 --> 00:00:23.430
the derivative of this, with
respect to x, this is equal to

00:00:23.430 --> 00:00:29.540
the partial of psi, with respect
to x, plus the partial

00:00:29.540 --> 00:00:35.400
of psi, with respect
to y, times dy, dx.

00:00:35.400 --> 00:00:37.630
And in the last video I didn't
prove it to you, but I

00:00:37.630 --> 00:00:40.260
hopefully gave you a little bit
of intuition that you can

00:00:40.260 --> 00:00:40.740
believe me.

00:00:40.740 --> 00:00:43.030
But maybe one day I'll prove
it a little bit more

00:00:43.030 --> 00:00:46.120
rigorously, but you can find
proofs on the web if you are

00:00:46.120 --> 00:00:49.960
interested, for the chain rule
with partial derivatives.

00:00:49.960 --> 00:00:52.760
So let's put that aside and
let's explore another property

00:00:52.760 --> 00:00:55.600
of partial derivatives, and then
we're ready to get the

00:00:55.600 --> 00:00:57.080
intuition behind exact
equations.

00:00:57.080 --> 00:00:59.070
Because you're going to find,
it's fairly straightforward to

00:00:59.070 --> 00:01:02.210
solve exact equations, but the
intuition is a little bit

00:01:02.210 --> 00:01:05.140
more-- well, I don't want to say
it's difficult, because if

00:01:05.140 --> 00:01:06.890
you have the intuition,
you have it.

00:01:06.890 --> 00:01:11.490
So what if I had, say, this
function, psi, and I were to

00:01:11.490 --> 00:01:16.580
take the partial derivative of
psi, with respect to x, first.

00:01:16.580 --> 00:01:17.510
I'll just write psi.

00:01:17.510 --> 00:01:19.640
I don't have to write
x and y every time.

00:01:19.640 --> 00:01:22.890
And then I were to take the
partial derivative with

00:01:22.890 --> 00:01:25.485
respect to y.

00:01:28.920 --> 00:01:32.730
So just as a notation, this you
could write as, you could

00:01:32.730 --> 00:01:34.620
kind of view it as you're
multiplying the operators, so

00:01:34.620 --> 00:01:36.050
it could be written like this.

00:01:36.050 --> 00:01:42.400
The partial del squared times
psi, or del squared psi, over

00:01:42.400 --> 00:01:47.540
del y del, or curly d x.

00:01:47.540 --> 00:01:50.330
And that can also be written
as-- and this is my preferred

00:01:50.330 --> 00:01:53.040
notation, because it doesn't
have all this extra junk

00:01:53.040 --> 00:01:53.800
everywhere.

00:01:53.800 --> 00:01:56.350
You could just say, well, the
partial, we took the partial,

00:01:56.350 --> 00:02:00.050
with respect to x, first. So
this just means the partial of

00:02:00.050 --> 00:02:01.240
psi, with respect to x.

00:02:01.240 --> 00:02:04.060
And then we took the partial,
with respect to y.

00:02:04.060 --> 00:02:05.870
So that's one situation
to consider.

00:02:05.870 --> 00:02:07.970
What happens when we take the
partial, with respect to x,

00:02:07.970 --> 00:02:08.650
and then y?

00:02:08.650 --> 00:02:13.100
So with respect to x, you hold
y constant to get just the

00:02:13.100 --> 00:02:14.190
partial, with respect to x.

00:02:14.190 --> 00:02:15.000
Ignore the y there.

00:02:15.000 --> 00:02:17.060
And then you hold the x
constant, and you take the

00:02:17.060 --> 00:02:18.670
partial, with respect to y.

00:02:18.670 --> 00:02:21.480
So what's the difference between
that and if we were to

00:02:21.480 --> 00:02:22.370
switch the order?

00:02:22.370 --> 00:02:24.970
So what happens if we were to--
I'll do it in a different

00:02:24.970 --> 00:02:30.400
color-- if we had psi, and we
were to take the partial, with

00:02:30.400 --> 00:02:34.480
respect to y, first, and then
we were to take the partial,

00:02:34.480 --> 00:02:36.510
with respect to x?

00:02:36.510 --> 00:02:40.640
So just the notation, just so
you're comfortable with it,

00:02:40.640 --> 00:02:44.660
that would be-- so partial
x, partial y.

00:02:44.660 --> 00:02:46.360
And this is the operator.

00:02:46.360 --> 00:02:48.750
And it might be a little
confusing that here, between

00:02:48.750 --> 00:02:51.060
these two notations, even though
they're the same thing,

00:02:51.060 --> 00:02:52.740
the order is mixed.

00:02:52.740 --> 00:02:54.250
That's just because it's
just a different way of

00:02:54.250 --> 00:02:54.910
thinking about it.

00:02:54.910 --> 00:02:57.990
This says, OK, partial first,
with respect to x, then y.

00:02:57.990 --> 00:03:00.160
This views it more as the
operator, so we took the

00:03:00.160 --> 00:03:03.000
partial of x first, and then
we took y, like you're

00:03:03.000 --> 00:03:04.950
multiplying the operators.

00:03:04.950 --> 00:03:08.840
But anyway, so this can also be
written as the partial of

00:03:08.840 --> 00:03:13.070
y, with respect to x-- sorry,
the partial of y, and then we

00:03:13.070 --> 00:03:14.910
took the partial of that
with respect to x.

00:03:14.910 --> 00:03:17.980
Now, I'm going to tell you right
now, that if each of the

00:03:17.980 --> 00:03:20.840
first partials are continuous--
and most of the

00:03:20.840 --> 00:03:24.510
functions we've dealt with in
a normal domain, as long as

00:03:24.510 --> 00:03:26.780
there aren't any
discontinuities, or holes, or

00:03:26.780 --> 00:03:29.070
something strange in the
function definition, they

00:03:29.070 --> 00:03:30.290
usually are continuous.

00:03:30.290 --> 00:03:32.990
And especially in a first-year
calculus or differential

00:03:32.990 --> 00:03:35.810
course, we're probably going to
be dealing with continuous

00:03:35.810 --> 00:03:37.620
functions in soon. our domain.

00:03:37.620 --> 00:03:40.480
If both of these functions are
continuous, if both of the

00:03:40.480 --> 00:03:45.410
first partials are continuous,
then these two are going to be

00:03:45.410 --> 00:03:47.170
equal to each other.

00:03:47.170 --> 00:03:54.950
So psi of xy is going to
be equal to psi of yx.

00:03:54.950 --> 00:04:01.220
Now, we can use this knowledge,
which is the chain

00:04:01.220 --> 00:04:04.870
rule using partial derivatives,
and this

00:04:04.870 --> 00:04:09.060
knowledge to now solve a certain
class of differential

00:04:09.060 --> 00:04:13.060
equations, first order
differential equations, called

00:04:13.060 --> 00:04:14.270
exact equations.

00:04:14.270 --> 00:04:17.860
And what does an exact
equation look like?

00:04:17.860 --> 00:04:21.990
An exact equation
looks like this.

00:04:21.990 --> 00:04:23.710
The color picking's
the hard part.

00:04:23.710 --> 00:04:26.290
So let's say this is my
differential equation.

00:04:26.290 --> 00:04:29.550
I have some function
of x and y.

00:04:29.550 --> 00:04:31.830
So I don't know, it could
be x squared times

00:04:31.830 --> 00:04:32.920
cosine of y or something.

00:04:32.920 --> 00:04:34.650
I don't know, it could be
any function of x and y.

00:04:34.650 --> 00:04:40.350
Plus some function of x and y,
we'll call that n, times dy,

00:04:40.350 --> 00:04:44.900
dx is equal to 0.

00:04:44.900 --> 00:04:47.520
This is-- well, I don't know if
it's an exact equation yet,

00:04:47.520 --> 00:04:50.880
but if you saw something of this
form, your first impulse

00:04:50.880 --> 00:04:52.990
should be, oh-- well, actually,
your very first

00:04:52.990 --> 00:04:54.500
impulse is, is this separable?

00:04:54.500 --> 00:04:56.180
And you should try to play
around with the algebra a

00:04:56.180 --> 00:04:57.620
little bit to see if it's
separable, because that's

00:04:57.620 --> 00:04:59.210
always the most straightforward
way.

00:04:59.210 --> 00:05:01.770
If it's not separable, but you
can still put it in this form,

00:05:01.770 --> 00:05:04.460
you say, hey, is it
an exact equation?

00:05:04.460 --> 00:05:06.340
And what's an exact equation?

00:05:06.340 --> 00:05:07.270
Well, look immediately.

00:05:07.270 --> 00:05:11.600
This pattern right here
looks an awful

00:05:11.600 --> 00:05:14.000
lot like this pattern.

00:05:14.000 --> 00:05:18.210
What if M was the partial of
psi, with respect to x?

00:05:18.210 --> 00:05:24.920
What if psi, with respect
to x, is equal to M?

00:05:24.920 --> 00:05:26.710
What if this was psi,
with respect to x?

00:05:26.710 --> 00:05:29.570
And what if this was psi,
with respect to y?

00:05:29.570 --> 00:05:32.500
So psi, with respect to
y, is equal to N.

00:05:32.500 --> 00:05:32.950
What if?

00:05:32.950 --> 00:05:34.670
I'm just saying, we don't
know for sure, right?

00:05:34.670 --> 00:05:37.500
If you just see this someplace
randomly, you won't know for

00:05:37.500 --> 00:05:40.200
sure that this is the partial
of, with respect to x of some

00:05:40.200 --> 00:05:43.060
function, and this is the
partial, with respect to y of

00:05:43.060 --> 00:05:43.830
some function.

00:05:43.830 --> 00:05:45.810
But we're just saying,
what if?

00:05:45.810 --> 00:05:49.650
If this were true, then we
could rewrite this as the

00:05:49.650 --> 00:05:52.870
partial of psi, with respect to
x, plus the partial of psi,

00:05:52.870 --> 00:05:58.680
with respect to y, times
dy, dx, equal to 0.

00:05:58.680 --> 00:06:02.050
And this right here, the left
side right there, that's the

00:06:02.050 --> 00:06:04.790
same thing as this, right?

00:06:04.790 --> 00:06:09.040
This is just the derivative of
psi, with respect to x, using

00:06:09.040 --> 00:06:10.940
the partial derivative
chain rule.

00:06:10.940 --> 00:06:12.710
So you could rewrite it.

00:06:12.710 --> 00:06:17.130
You could rewrite, this is just
the derivative of psi,

00:06:17.130 --> 00:06:20.480
with respect to x, inside
the function of x,

00:06:20.480 --> 00:06:23.410
y, is equal to 0.

00:06:23.410 --> 00:06:27.730
So if you see a differential
equation, and it has this

00:06:27.730 --> 00:06:31.070
form, and you say, boy, I can't
separate it, but maybe

00:06:31.070 --> 00:06:32.030
it's an exact equation.

00:06:32.030 --> 00:06:35.940
And frankly, if that was what
was recently covered before

00:06:35.940 --> 00:06:38.800
the current exam, it probably
is an exact equation.

00:06:38.800 --> 00:06:40.940
But if you see this form,
you say, boy, maybe

00:06:40.940 --> 00:06:42.070
it's an exact equation.

00:06:42.070 --> 00:06:44.580
If it is an exact equation-- and
I'll show you how to test

00:06:44.580 --> 00:06:48.350
it in a second using this
information-- then this can be

00:06:48.350 --> 00:06:52.550
written as the derivative of
some function, psi, where this

00:06:52.550 --> 00:06:54.840
is the partial of psi,
with respect to x.

00:06:54.840 --> 00:06:57.720
This is the partial of psi,
with respect to y.

00:06:57.720 --> 00:06:59.655
And then if you could write it
like this, and you take the

00:06:59.655 --> 00:07:01.370
derivative of both sides--
sorry, you take the

00:07:01.370 --> 00:07:06.890
antiderivative of both sides--
and you would get psi of x, y

00:07:06.890 --> 00:07:10.070
is equal to c as a solution.

00:07:10.070 --> 00:07:12.770
So there are two things that we
should be caring you about.

00:07:12.770 --> 00:07:16.470
Then you might be saying, OK,
Sal, you've walked through

00:07:16.470 --> 00:07:19.550
psi's, and partials,
and all this.

00:07:19.550 --> 00:07:22.020
One, how do I know that it's
an exact equation?

00:07:22.020 --> 00:07:24.590
And then, if it is an exact
equation, which tells us that

00:07:24.590 --> 00:07:28.290
there is some psi, then how
do I solve for the psi?

00:07:28.290 --> 00:07:32.380
So the way to figure out is it
an exact equation, is to use

00:07:32.380 --> 00:07:34.690
this information right here.

00:07:34.690 --> 00:07:38.150
We know that if psi and its
derivatives are continuous

00:07:38.150 --> 00:07:42.100
over some domain, that when
you take the partial, with

00:07:42.100 --> 00:07:45.760
respect to x and then y, that's
the same thing as doing

00:07:45.760 --> 00:07:46.980
it in the other order.

00:07:46.980 --> 00:07:48.930
So we said, this is
the partial, with

00:07:48.930 --> 00:07:50.180
respect to x, right?

00:07:52.610 --> 00:07:55.920
And this is the partial,
with respect to y.

00:07:55.920 --> 00:07:59.880
So if this is an exact equation,
if this is the exact

00:07:59.880 --> 00:08:03.250
equation, if we were take the
partial of this, with respect

00:08:03.250 --> 00:08:05.330
to y, right?

00:08:05.330 --> 00:08:11.600
If we were to take the partial
of M, with respect to y-- so

00:08:11.600 --> 00:08:15.560
the partial of psi, with respect
to x, is equal to M.

00:08:15.560 --> 00:08:18.490
If we were to take the partial
of those, with respect to y--

00:08:18.490 --> 00:08:22.450
so we could just rewrite that as
that-- then that should be

00:08:22.450 --> 00:08:28.090
equal to the partial of N,
with respect to x, right?

00:08:28.090 --> 00:08:31.976
The partial of psi, with respect
to y, is equal to N.

00:08:31.976 --> 00:08:34.760
So if we take the partial, with
respect to x, of both of

00:08:34.760 --> 00:08:40.964
these, we know from this that
these should be equal, if psi

00:08:40.964 --> 00:08:44.400
and its partials are continuous
over that domain.

00:08:44.400 --> 00:08:49.320
So then this will
also be equal.

00:08:49.320 --> 00:08:51.990
So that is actually the
test to test if

00:08:51.990 --> 00:08:53.930
this is an exact equation.

00:08:53.930 --> 00:08:56.300
So let me rewrite all of that
again and summarize it a

00:08:56.300 --> 00:08:56.690
little bit.

00:08:56.690 --> 00:09:04.870
So if you see something of the
form, M of x, y plus N of x,

00:09:04.870 --> 00:09:09.580
y, times dy, dx is equal to 0.

00:09:09.580 --> 00:09:13.110
And then you take the partial
derivative of M, with respect

00:09:13.110 --> 00:09:18.280
to y, and then you take the
partial derivative of N, with

00:09:18.280 --> 00:09:24.030
respect to x, and they are equal
to each other, then--

00:09:24.030 --> 00:09:26.410
and it's actually if and only
if, so it goes both ways--

00:09:26.410 --> 00:09:30.930
this is an exact equation, an
exact differential equation.

00:09:30.930 --> 00:09:32.410
This is an exact equation.

00:09:32.410 --> 00:09:35.510
And if it's an exact equation,
that tells us that there

00:09:35.510 --> 00:09:47.140
exists a psi, such that the
derivative of psi of x, y is

00:09:47.140 --> 00:09:52.200
equal to 0, or psi of x, y is
equal to c, is a solution of

00:09:52.200 --> 00:09:53.050
this equation.

00:09:53.050 --> 00:09:58.480
And the partial derivative of
psi, with respect to x, is

00:09:58.480 --> 00:09:59.740
equal to M.

00:09:59.740 --> 00:10:03.760
And the partial derivative of
psi, with respect to y, is

00:10:03.760 --> 00:10:05.340
equal to N.

00:10:05.340 --> 00:10:07.550
And I'll show you in the next
video how to actually use this

00:10:07.550 --> 00:10:09.810
information to solve for psi.

00:10:09.810 --> 00:10:11.640
So here are some things
I want to point out.

00:10:11.640 --> 00:10:13.720
This is going to be the partial
derivative of psi,

00:10:13.720 --> 00:10:17.620
with respect to x, but when we
take the kind of exact test,

00:10:17.620 --> 00:10:19.590
we take it with respect to y,
because we want to get that

00:10:19.590 --> 00:10:21.080
mixed derivative.

00:10:21.080 --> 00:10:23.410
Similarly, this is going to be
the partial derivative of psi,

00:10:23.410 --> 00:10:27.030
with respect to y, but when we
do the test, we take the

00:10:27.030 --> 00:10:29.500
partial of it with respect
to x so we get that mixed

00:10:29.500 --> 00:10:30.730
derivative.

00:10:30.730 --> 00:10:32.570
This is with respect to y,
and then with respect to

00:10:32.570 --> 00:10:33.920
x, so you get this.

00:10:33.920 --> 00:10:36.300
Anyway, I know that might be a
little bit involved, but if

00:10:36.300 --> 00:10:38.360
you understood everything I did,
I think you'll have the

00:10:38.360 --> 00:10:41.390
intuition behind why
the methodology of

00:10:41.390 --> 00:10:43.470
exact equations works.

00:10:43.470 --> 00:10:45.950
I will see you in the next
video, where we will actually

00:10:45.950 --> 00:10:49.400
solve some exact equations See


Exact equations example 1

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:03.600
OK, I filled your brain with a
bunch of partial derivatives

00:00:03.600 --> 00:00:05.920
and psi's, with respect
to x's and y's.

00:00:05.920 --> 00:00:09.820
I think now it's time to
actually do it with a real

00:00:09.820 --> 00:00:11.400
differential equation,
and make things a

00:00:11.400 --> 00:00:12.195
little bit more concrete.

00:00:12.195 --> 00:00:16.640
So let's say I have the
differential, y, the

00:00:16.640 --> 00:00:24.570
differential equation, y cosine
of x, plus 2xe to the

00:00:24.570 --> 00:00:34.360
y, plus sine of x, plus-- I'm
already running out of space--

00:00:34.360 --> 00:00:40.140
x squared, e to the y,
minus 1, times y

00:00:40.140 --> 00:00:42.970
prime, is equal to 0.

00:00:42.970 --> 00:00:46.090
Well, your brain is already,
hopefully, in exact

00:00:46.090 --> 00:00:47.310
differential equations mode.

00:00:47.310 --> 00:00:49.510
But if you were to see this
pattern in general, where you

00:00:49.510 --> 00:00:53.690
see a function of x and y,
here-- this is just some

00:00:53.690 --> 00:00:56.870
function of x and y-- and then
you have another function of x

00:00:56.870 --> 00:01:00.440
and y, times y prime, or times
dy, d of x, your brain should

00:01:00.440 --> 00:01:01.950
immediately say if this
is inseparable.

00:01:01.950 --> 00:01:04.180
And I'm not going to try to
make it separable, just

00:01:04.180 --> 00:01:05.530
because that'll take
a lot of time.

00:01:05.530 --> 00:01:07.460
But if it's not separable, your
brain said, oh, maybe

00:01:07.460 --> 00:01:08.850
this is an exact equation.

00:01:08.850 --> 00:01:11.350
And, you say, let
me test whether

00:01:11.350 --> 00:01:13.040
this is an exact equation.

00:01:13.040 --> 00:01:16.300
So if this is an exact equation,
this is our function

00:01:16.300 --> 00:01:19.160
M, which is a function
of x and y.

00:01:19.160 --> 00:01:21.670
And this is our function
N, which is a

00:01:21.670 --> 00:01:23.140
function of x and y.

00:01:23.140 --> 00:01:25.700
Now, the test is to see if the
partial of this, with respect

00:01:25.700 --> 00:01:30.290
to y, is equal to the partial
of this, with respect to x.

00:01:30.290 --> 00:01:31.420
So let's see.

00:01:31.420 --> 00:01:36.240
The partial of M, with respect
to y, is equal to-- let's see,

00:01:36.240 --> 00:01:39.480
y is-- so this cosine of x is
just a constant, so it's just

00:01:39.480 --> 00:01:42.460
cosine of x.

00:01:42.460 --> 00:01:47.610
Cosine of x plus-- now,
what's the derivative?

00:01:47.610 --> 00:01:49.725
Well, 2x is just a constant,
what's the derivative of e to

00:01:49.725 --> 00:01:50.820
the y, with respect to y?

00:01:50.820 --> 00:01:53.350
Well, it's just e
to the y, right?

00:01:53.350 --> 00:01:56.110
So we have the constant on
the outside, 2x times the

00:01:56.110 --> 00:01:58.810
derivative, with respect to
y, so it's 2xe to the y.

00:01:58.810 --> 00:01:59.910
Fair enough.

00:01:59.910 --> 00:02:01.430
Now, what is the partial
derivative of this, with

00:02:01.430 --> 00:02:02.950
respect to x?

00:02:02.950 --> 00:02:09.100
So N sub x, or the partial of
N, with respect to x-- so

00:02:09.100 --> 00:02:11.360
what's the derivative of sine
of x, with respect to x?

00:02:11.360 --> 00:02:17.630
Well, that's easy, that's cosine
of x, plus 2x times e

00:02:17.630 --> 00:02:19.830
to the y, right? e and y is just
a constant, because y is

00:02:19.830 --> 00:02:22.030
constant when we're taking the
partial, with respect to x.

00:02:22.030 --> 00:02:26.550
So plus 2xe to the y.

00:02:26.550 --> 00:02:28.800
And then minus 1, the derivative
of a constant, with

00:02:28.800 --> 00:02:31.450
respect to anything
is going to be 0.

00:02:31.450 --> 00:02:34.310
So the derivative of N-- the
partial of N, with respect to

00:02:34.310 --> 00:02:38.460
x, is cosine of x, plus 2xe to
the y, which, lo and behold,

00:02:38.460 --> 00:02:41.000
is the same thing as the
derivative, the partial of M,

00:02:41.000 --> 00:02:43.120
with respect to y.

00:02:43.120 --> 00:02:44.190
So there we have it.

00:02:44.190 --> 00:02:48.860
We've shown that M of y is equal
to-- or the partial of

00:02:48.860 --> 00:02:52.230
M, with respect to y-- is equal
to the partial of N,

00:02:52.230 --> 00:02:55.510
with respect to x, which
tells us that

00:02:55.510 --> 00:02:59.040
this is an exact equation.

00:02:59.040 --> 00:03:06.380
Now, given that this is an exact
equation-- oh, my wife

00:03:06.380 --> 00:03:08.120
snuck up behind me,
I was wondering.

00:03:08.120 --> 00:03:09.150
I thought there was
some critter in

00:03:09.150 --> 00:03:09.820
my house, or something.

00:03:09.820 --> 00:03:13.440
Anyway, so we know that this is
an exact equation, so what

00:03:13.440 --> 00:03:14.020
does that tell us?

00:03:14.020 --> 00:03:17.990
Well, that tells us that there's
some psi, where the

00:03:17.990 --> 00:03:22.060
partial derivative of psi, with
respect to x, is equal to

00:03:22.060 --> 00:03:26.420
M, and the partial derivative of
psi, with respect to y, is

00:03:26.420 --> 00:03:27.600
equal to N.

00:03:27.600 --> 00:03:30.800
And if we know that psi,
then we can rewrite our

00:03:30.800 --> 00:03:35.090
differential equation as the
derivative of psi, with

00:03:35.090 --> 00:03:38.450
respect to x, is equal to 0.

00:03:38.450 --> 00:03:40.790
So let's solve for psi.

00:03:40.790 --> 00:03:43.765
So we know that the partial of
psi, with respect to x, is

00:03:43.765 --> 00:03:44.390
equal to M.

00:03:44.390 --> 00:03:45.160
So we could write that.

00:03:45.160 --> 00:03:50.820
We could write the partial of
psi, with respect to x, is

00:03:50.820 --> 00:04:01.110
equal to M, which is y cosine
of x, plus 2xe to the y.

00:04:01.110 --> 00:04:01.600
That's just here.

00:04:01.600 --> 00:04:02.570
That's my M of x.

00:04:02.570 --> 00:04:03.440
We could have done
it the other way.

00:04:03.440 --> 00:04:06.160
We could have said the partial
of y-- the partial of psi,

00:04:06.160 --> 00:04:08.220
with respect to y, is this
thing over here.

00:04:08.220 --> 00:04:10.430
But let's just do it with x.

00:04:10.430 --> 00:04:14.910
Now, to at least get kind of a
first approximation of what

00:04:14.910 --> 00:04:16.930
size-- not an approximation, but
to start to get a sense of

00:04:16.930 --> 00:04:19.829
it-- let's take the derivative
of both sides, with respect

00:04:19.829 --> 00:04:22.000
to-- sorry, take the
antiderivative-- take the

00:04:22.000 --> 00:04:25.270
integral of both sides,
with respect to x.

00:04:25.270 --> 00:04:28.280
So if you take the derivative of
this, with respect to x, if

00:04:28.280 --> 00:04:30.800
you integrate-- sorry, if
you were to take the

00:04:30.800 --> 00:04:34.220
antiderivative of this,
with respect to x.

00:04:34.220 --> 00:04:36.836
So let me just write
that down.

00:04:36.836 --> 00:04:38.520
The partial, with
respect to x.

00:04:38.520 --> 00:04:39.360
We're going to [? integrate ?]

00:04:39.360 --> 00:04:40.820
it, with respect to x.

00:04:40.820 --> 00:04:45.030
That is going to be equal to
the integral of this whole

00:04:45.030 --> 00:04:46.820
thing, with respect to x.

00:04:46.820 --> 00:04:51.940
Cosine of x plus 2xe to the y.

00:04:51.940 --> 00:04:53.870
We're integrating with
respect to x.

00:04:53.870 --> 00:04:57.220
And normally when you integrate
with respect to x,

00:04:57.220 --> 00:04:59.430
you'd say, OK, plus c, right?

00:04:59.430 --> 00:05:03.000
But it actually could be a
plus-- since this is a

00:05:03.000 --> 00:05:05.740
partial, with respect to x, we
could have had some function

00:05:05.740 --> 00:05:09.045
of y here in general, because
y, we treat it

00:05:09.045 --> 00:05:10.730
as a constant, right?

00:05:10.730 --> 00:05:12.300
And that makes sense, because
if you were to take the

00:05:12.300 --> 00:05:15.110
partial of both sides of this,
with respect to x, if you were

00:05:15.110 --> 00:05:17.430
to take the partial of a
function that is only a

00:05:17.430 --> 00:05:19.420
function of y, with respect
to x, you would

00:05:19.420 --> 00:05:20.880
have gotten a 0 here.

00:05:20.880 --> 00:05:22.200
So when you take the
antiderivative, you were,

00:05:22.200 --> 00:05:24.190
like, oh well, there might have
been some function of y

00:05:24.190 --> 00:05:26.780
here that we lost when
we took the partial,

00:05:26.780 --> 00:05:28.790
with respect to x.

00:05:28.790 --> 00:05:34.690
So anyway, this will
simplify to psi.

00:05:34.690 --> 00:05:37.710
psi is going to be equal to the
integral, with respect to

00:05:37.710 --> 00:05:40.340
x, or the antiderivative, with
respect to x, here, plus some

00:05:40.340 --> 00:05:44.130
function of y that we might
have lost when we took the

00:05:44.130 --> 00:05:46.520
partial, with respect to x.

00:05:46.520 --> 00:05:47.260
So let's do that.

00:05:47.260 --> 00:05:49.260
Let's figure out
this integral.

00:05:49.260 --> 00:05:52.890
I'll do it in blue So y
is just a constant.

00:05:52.890 --> 00:05:57.310
So the antiderivative of y
cosine of x, is just y sine of

00:05:57.310 --> 00:06:04.020
x, plus-- either the y
is constant, so 2x.

00:06:04.020 --> 00:06:06.060
The antiderivative of 2x, with
respect to x, is x squared, so

00:06:06.060 --> 00:06:10.920
it's x squared e to the y.

00:06:10.920 --> 00:06:18.440
And then plus some
function of y.

00:06:18.440 --> 00:06:20.540
And if you want to verify this,
you should take the

00:06:20.540 --> 00:06:22.080
partial of this, with
respect to x.

00:06:22.080 --> 00:06:24.200
If you take the partial of
this, with respect to x,

00:06:24.200 --> 00:06:27.780
you're going to get this in
here, which is our function,

00:06:27.780 --> 00:06:30.310
M, up here.

00:06:30.310 --> 00:06:32.190
And then when you take the
partial of this, with respect

00:06:32.190 --> 00:06:34.720
to x, you'll get 0, and
it'll get lost. OK, so

00:06:34.720 --> 00:06:35.430
we're almost there.

00:06:35.430 --> 00:06:38.270
We've almost figured out our
psi, but we still need to

00:06:38.270 --> 00:06:42.030
figure out this function of y.

00:06:42.030 --> 00:06:43.880
Well, we know that if we take
the partial of this, with

00:06:43.880 --> 00:06:47.340
respect to y, since this
is an exact equation,

00:06:47.340 --> 00:06:48.210
we should get this.

00:06:48.210 --> 00:06:49.410
We should get our N function.

00:06:49.410 --> 00:06:51.150
So let's do that.

00:06:51.150 --> 00:06:55.540
So the partial-- I'll switch
notation, just to expose you

00:06:55.540 --> 00:06:58.340
to it-- the partial psi, with
respect to y, is going to be

00:06:58.340 --> 00:07:02.160
equal to-- so here, y sine
of x, sine of x is just a

00:07:02.160 --> 00:07:04.420
constant. y is just y, so the
derivative of this, with

00:07:04.420 --> 00:07:06.030
respect to y, is
just sine of x.

00:07:08.920 --> 00:07:12.300
Plus the derivative of e to the
y is e to the y. x squared

00:07:12.300 --> 00:07:12.950
is just a constant.

00:07:12.950 --> 00:07:18.600
So it's just x squared e to
the y, plus-- what's the

00:07:18.600 --> 00:07:21.120
partial of f of y, with
respect to y?

00:07:21.120 --> 00:07:26.900
It's going to be f prime of y.

00:07:26.900 --> 00:07:27.530
Well, what did we do?

00:07:27.530 --> 00:07:31.230
We took M, we integrated with
respect to x, and we said,

00:07:31.230 --> 00:07:33.270
well, we might have lost some
function of y, so we added

00:07:33.270 --> 00:07:34.060
that to it.

00:07:34.060 --> 00:07:37.740
And then we took the partial of
that side that we've almost

00:07:37.740 --> 00:07:40.470
constructed, and we took
the partial of that,

00:07:40.470 --> 00:07:42.040
with respect to y.

00:07:42.040 --> 00:07:44.940
Now, we know, since this is
exact, that that is going to

00:07:44.940 --> 00:07:46.960
equal our N.

00:07:46.960 --> 00:07:48.670
So our N is up there.

00:07:48.670 --> 00:07:52.585
Cosine of x plus-- So that's
going to be equal to-- I want

00:07:52.585 --> 00:07:57.720
to make sure I can read it up
there-- to our N, right?

00:07:57.720 --> 00:07:58.390
Oh no, sorry.

00:07:58.390 --> 00:07:59.930
N is up here.

00:07:59.930 --> 00:08:00.540
Our N is up here.

00:08:00.540 --> 00:08:03.150
Sine of x-- let me write
that-- sine of x plus x

00:08:03.150 --> 00:08:05.450
squared, e to the y, minus 1.

00:08:05.450 --> 00:08:13.760
So sine of x plus x squared,
e to the y, minus 1.

00:08:17.820 --> 00:08:19.350
That was just our N,
from our original

00:08:19.350 --> 00:08:21.470
differential equation.

00:08:21.470 --> 00:08:23.720
And now we can solve
for f prime of y.

00:08:23.720 --> 00:08:31.010
So let's see, we get sine of x
plus x squared, e to the y,

00:08:31.010 --> 00:08:39.364
plus f prime of y, is equal to
sine of x plus x squared, e to

00:08:39.364 --> 00:08:42.150
the y, minus 1.

00:08:42.150 --> 00:08:45.600
So let's see, we can delete
sine of x from both sides.

00:08:45.600 --> 00:08:48.640
We can delete x squared e to
the y from both sides.

00:08:48.640 --> 00:08:49.740
And then what are
we left with?

00:08:49.740 --> 00:08:55.750
We're left with f prime
of y is equal to 1.

00:08:55.750 --> 00:09:03.610
And then we're left with f of
y is equal to-- well, it

00:09:03.610 --> 00:09:11.720
equals y plus some constant,
c, right?

00:09:11.720 --> 00:09:14.350
So what is our psi now?

00:09:14.350 --> 00:09:17.040
We wrote our psi up here, and we
had this f of y here, so we

00:09:17.040 --> 00:09:17.600
can rewrite it now.

00:09:17.600 --> 00:09:20.640
So psi is a function of x and
y-- we're actually pretty much

00:09:20.640 --> 00:09:24.240
almost done solving it-- psi
is a function of x and y is

00:09:24.240 --> 00:09:41.010
equal to y sine of x, plus x
squared, e to the y, plus y--

00:09:41.010 --> 00:09:42.860
oh, sorry, this is f prime
of y, minus 1.

00:09:42.860 --> 00:09:44.330
So this is a minus 1.

00:09:44.330 --> 00:09:46.350
So this is a minus y plus c.

00:09:46.350 --> 00:09:49.370
So this is going to be
a minus y plus c.

00:09:49.370 --> 00:09:51.560
So we've solved for psi.

00:09:51.560 --> 00:09:52.800
And so what does that tell us?

00:09:52.800 --> 00:09:55.390
Well, we said that original
differential equation, up

00:09:55.390 --> 00:09:59.390
here, using the partial
derivative chain rule, that

00:09:59.390 --> 00:10:03.270
original differential equation,
can be rewritten now

00:10:03.270 --> 00:10:10.740
as the derivative dx of psi is
equal to-- psi is a function

00:10:10.740 --> 00:10:14.570
of x and y-- is equal to 0.

00:10:14.570 --> 00:10:17.560
Or if you were to integrate both
sides of this, you would

00:10:17.560 --> 00:10:26.630
get that psi of xy is equal
to c is a solution of that

00:10:26.630 --> 00:10:27.430
differential equation.

00:10:27.430 --> 00:10:30.020
So if we were to set this is
equal to c, that's the

00:10:30.020 --> 00:10:30.700
differential equation.

00:10:30.700 --> 00:10:37.530
So we could say, y sine of x
plus x squared, e to the y,

00:10:37.530 --> 00:10:42.890
minus y-- now we could say, plus
this c-- plus this c, you

00:10:42.890 --> 00:10:44.990
call it c1, is equal to c2.

00:10:44.990 --> 00:10:47.143
Well, you could subtract the c's
from both sides, and just

00:10:47.143 --> 00:10:48.520
be left with a c at the end.

00:10:48.520 --> 00:10:53.380
But anyway, we have solved
this exact equation, one,

00:10:53.380 --> 00:10:57.220
first, by recognizing it was
exact, by taking the partial

00:10:57.220 --> 00:11:01.670
of this, with respect to y, and
seeing if that was equal

00:11:01.670 --> 00:11:04.110
to the partial of N,
with respect to x.

00:11:04.110 --> 00:11:07.000
Once we saw that they were
equal, we're like, OK, this is

00:11:07.000 --> 00:11:08.090
going to be exact.

00:11:08.090 --> 00:11:09.850
So let's figure out psi.

00:11:09.850 --> 00:11:13.340
Since this is exact, M is going
to be the partial of

00:11:13.340 --> 00:11:15.000
psi, with respect to x.

00:11:15.000 --> 00:11:17.700
N is the partial of psi,
with respect to y.

00:11:17.700 --> 00:11:21.570
Then to figure out y, we
integrated M, with respect to

00:11:21.570 --> 00:11:25.130
x, and we got this.

00:11:25.130 --> 00:11:27.500
But since we said, oh, well,
instead of a plus c, it could

00:11:27.500 --> 00:11:29.520
have been a function of y there,
because we took the

00:11:29.520 --> 00:11:32.140
partial, with respect to x, so
this might have been lost. To

00:11:32.140 --> 00:11:36.100
figure out the function of y,
we then took our psi that we

00:11:36.100 --> 00:11:37.960
figured out, took the partial
of that, with

00:11:37.960 --> 00:11:40.290
respect to y, got this.

00:11:40.290 --> 00:11:42.450
And we said, this was an exact
equation, so this is going to

00:11:42.450 --> 00:11:45.710
equal our N of x y.

00:11:45.710 --> 00:11:47.750
We set those equal to each
other, and then we

00:11:47.750 --> 00:11:50.420
solved for f of y.

00:11:50.420 --> 00:11:52.690
And then we had our final psi.

00:11:52.690 --> 00:11:54.320
Our final psi was this.

00:11:54.320 --> 00:11:58.500
And then the differential
equation, because of the chain

00:11:58.500 --> 00:12:01.270
rule of partial derivatives,
we could rewrite the

00:12:01.270 --> 00:12:02.330
differential equation as this.

00:12:02.330 --> 00:12:05.000
The solution is this, and so
this is the solution to our

00:12:05.000 --> 00:12:06.510
differential equation.

00:12:06.510 --> 00:12:08.780
See you in the next video.


Exact equations intuition 1 (proofy)

WEBVTT
Kind: captions
Language: en

00:00:00.890 --> 00:00:04.650
Now I introduce you to the
concept of exact equations.

00:00:04.650 --> 00:00:07.470
And it's just another method for
solving a certain type of

00:00:07.470 --> 00:00:08.370
differential equations.

00:00:08.370 --> 00:00:09.220
Let me write that down.

00:00:09.220 --> 00:00:12.610
Exact equations.

00:00:15.620 --> 00:00:18.710
Before I show you what an exact
equation is, I'm just

00:00:18.710 --> 00:00:21.050
going to give you a little bit
of the building blocks, just

00:00:21.050 --> 00:00:22.970
so that when I later prove it,
or at least give you the

00:00:22.970 --> 00:00:26.260
intuition behind it, it doesn't
seem like it's coming

00:00:26.260 --> 00:00:27.080
out of the blue.

00:00:27.080 --> 00:00:30.500
So let's say I had some function
of x and y, and we'll

00:00:30.500 --> 00:00:33.430
call it psi, just because that's
what people tend to use

00:00:33.430 --> 00:00:35.080
for these exact equations.

00:00:35.080 --> 00:00:37.220
So psi is a function
of x and y.

00:00:41.280 --> 00:00:47.280
So you're probably not familiar
with taking the chain

00:00:47.280 --> 00:00:49.870
rule onto partial derivatives,
but I'll show it to you now,

00:00:49.870 --> 00:00:51.030
and I'll give you a little
intuition, although

00:00:51.030 --> 00:00:51.950
I won't prove it.

00:00:51.950 --> 00:00:54.210
So if I were to take the
derivative of this with

00:00:54.210 --> 00:00:59.010
respect to x, where y is also
function of x, I could also

00:00:59.010 --> 00:01:04.709
write this as y-- sorry,
it's not y, psi.

00:01:04.709 --> 00:01:06.130
Undo.

00:01:06.130 --> 00:01:10.890
So I could also write this as
psi, as x and y, which is a

00:01:10.890 --> 00:01:11.760
function of x.

00:01:11.760 --> 00:01:12.820
I could write it
just like that.

00:01:12.820 --> 00:01:13.940
These are just two
different ways of

00:01:13.940 --> 00:01:14.990
writing the same thing.

00:01:14.990 --> 00:01:18.440
Now, if I were to take the
derivative of psi with respect

00:01:18.440 --> 00:01:23.100
to x-- and these are just the
building blocks-- if I were to

00:01:23.100 --> 00:01:28.020
take the derivative of psi with
respect to x, it is equal

00:01:28.020 --> 00:01:30.410
to-- this is the chain rule
using partial derivatives.

00:01:30.410 --> 00:01:31.820
And I won't prove it, but
I'll give you the

00:01:31.820 --> 00:01:33.390
intuition right here.

00:01:33.390 --> 00:01:37.410
So this is going to be equal to
the partial derivative of

00:01:37.410 --> 00:01:45.980
psi with respect to x plus the
partial derivative of psi with

00:01:45.980 --> 00:01:49.780
respect to y times dy dx.

00:01:53.300 --> 00:01:55.720
And this is should make a
little bit of intuition.

00:01:55.720 --> 00:01:58.470
I'm kind of taking the
derivative with respect to x,

00:01:58.470 --> 00:02:02.410
and if you could say, and I know
you can't, because this

00:02:02.410 --> 00:02:04.190
partial with respect to
y, and the dy, they're

00:02:04.190 --> 00:02:04.860
two different things.

00:02:04.860 --> 00:02:06.540
But if these canceled out,
then you'd kind of have

00:02:06.540 --> 00:02:08.590
another partial with
respect to x.

00:02:08.590 --> 00:02:12.030
And if you were to kind of add
them up, then you would get

00:02:12.030 --> 00:02:14.040
the full derivative
with respect to x.

00:02:14.040 --> 00:02:16.600
That's not even the intuition,
that's just to show you that

00:02:16.600 --> 00:02:19.970
even this should make a little
bit of intuitive sense.

00:02:19.970 --> 00:02:24.280
Now the intuition here, let's
just say psi, and psi doesn't

00:02:24.280 --> 00:02:27.380
always have to take this form,
but you could use this same

00:02:27.380 --> 00:02:31.600
methodology to take psi to
more complex notations.

00:02:31.600 --> 00:02:35.560
But let's say that psi, and
I won't write that it's a

00:02:35.560 --> 00:02:36.300
function of x and y.

00:02:36.300 --> 00:02:37.320
We know that it's a function
of x and y.

00:02:37.320 --> 00:02:43.190
Let's say it's equal to some
function of x, we'll call that

00:02:43.190 --> 00:02:48.480
f1 of x, times some
function of y.

00:02:48.480 --> 00:02:51.050
And let's say there's a bunch
of terms like this.

00:02:51.050 --> 00:02:54.780
So there's n terms like this,
plus all the way to the nth

00:02:54.780 --> 00:03:02.660
term is the nth function of x
times the nth function of y.

00:03:02.660 --> 00:03:05.570
I just defined psi like this
just so I can give you the

00:03:05.570 --> 00:03:09.380
intuition that when I use
implicit differentiation on

00:03:09.380 --> 00:03:11.000
this, when I take the derivative
of this with

00:03:11.000 --> 00:03:12.450
respect to x, I actually
get something that

00:03:12.450 --> 00:03:13.850
looks just like that.

00:03:13.850 --> 00:03:16.260
So what's the derivative of
psi with respect to x?

00:03:21.190 --> 00:03:23.700
And this is just the implicit
differentiation that you

00:03:23.700 --> 00:03:26.780
learned, or that you hopefully
learned, in your first

00:03:26.780 --> 00:03:28.640
semester calculus course.

00:03:28.640 --> 00:03:31.150
That's equal, and we just do
the product rule, right?

00:03:31.150 --> 00:03:33.500
So the first expression, you
take the derivative of that

00:03:33.500 --> 00:03:34.540
with respect to x.

00:03:34.540 --> 00:03:41.970
Well, that's just going to be f1
prime of x times the second

00:03:41.970 --> 00:03:46.650
function, well, that's
just g1 of y.

00:03:46.650 --> 00:03:50.640
Now you add that to the
derivative of the second

00:03:50.640 --> 00:03:53.690
function times the
first function.

00:03:53.690 --> 00:03:58.500
So plus f1 of x, that's just the
first function, times the

00:03:58.500 --> 00:03:59.690
derivative of the
second function.

00:03:59.690 --> 00:04:02.280
Now the derivative of the second
function, it's going to

00:04:02.280 --> 00:04:04.030
be this function with
respect to y.

00:04:04.030 --> 00:04:09.015
So you could write that
as g1 prime of y.

00:04:09.015 --> 00:04:11.050
But of course, we're doing
the chain rule.

00:04:11.050 --> 00:04:12.930
So it's that times dy dx.

00:04:12.930 --> 00:04:17.149
And you might want to review the
implicit differentiation

00:04:17.149 --> 00:04:19.399
videos if that seems a
little bit foreign.

00:04:19.399 --> 00:04:23.430
But this right here, what I
just did, this expression

00:04:23.430 --> 00:04:26.070
right here, this is the
derivative with

00:04:26.070 --> 00:04:28.570
respect to x of this.

00:04:28.570 --> 00:04:30.040
And we have n terms like that.

00:04:30.040 --> 00:04:33.170
So if we keep adding them, I'll
do them vertically down.

00:04:33.170 --> 00:04:36.630
So plus, and then you have a
bunch of them, and the last

00:04:36.630 --> 00:04:38.320
one's going to look the
same, it's just the

00:04:38.320 --> 00:04:40.660
nth function of x.

00:04:40.660 --> 00:04:49.400
So fn prime of x times the
second function, g n of y,

00:04:49.400 --> 00:04:55.930
plus the first function, fn of
x, times the derivative of the

00:04:55.930 --> 00:04:56.580
second function.

00:04:56.580 --> 00:04:59.350
The derivative of the second
function with respect to y is

00:04:59.350 --> 00:05:04.980
just g prime of y times dy dx.

00:05:04.980 --> 00:05:06.650
It's just a chain rule.

00:05:06.650 --> 00:05:09.620
dy dx.

00:05:09.620 --> 00:05:13.440
Now, we have two n terms. We
have n terms here, right,

00:05:13.440 --> 00:05:17.260
where each term was a f of x
times a g of y, or f1 of x

00:05:17.260 --> 00:05:19.040
times g1 of y, and then
all the way to fn of

00:05:19.040 --> 00:05:20.590
x times gn of y.

00:05:20.590 --> 00:05:23.150
Now for each of those, we got
two of them when we did the

00:05:23.150 --> 00:05:25.220
product rule.

00:05:25.220 --> 00:05:28.850
If we group the terms, so if
we group all the terms that

00:05:28.850 --> 00:05:31.500
don't have a dy dx on them,
what do we get?

00:05:31.500 --> 00:05:33.430
If we add all of these, I guess
you could call them on

00:05:33.430 --> 00:05:42.010
the left hand side, I'm just
rearranging, it all equals f1

00:05:42.010 --> 00:05:54.660
prime of x times g1 of y, plus
f2, g2, all the way to fn

00:05:54.660 --> 00:06:00.770
prime, I'm sorry, fn prime
of x, gn of y.

00:06:00.770 --> 00:06:06.900
That's just all of these
added up, plus all

00:06:06.900 --> 00:06:07.890
of these added up.

00:06:07.890 --> 00:06:13.080
All the terms that have
the dy dx in them.

00:06:13.080 --> 00:06:15.640
And I'll do them in
a different color.

00:06:15.640 --> 00:06:19.080
So all of these terms
are going to be

00:06:19.080 --> 00:06:20.092
in a different color.

00:06:20.092 --> 00:06:21.880
I'll do it in a different
parentheses.

00:06:21.880 --> 00:06:30.990
Plus f1 of x g1 prime of y, and
I'll do the dy dx later,

00:06:30.990 --> 00:06:32.100
I'll distribute it out.

00:06:32.100 --> 00:06:41.370
Plus, and we have n terms, plus
fn of x gn prime of y,

00:06:41.370 --> 00:06:47.200
and then all of these terms
are multiplied by dy dx.

00:06:47.200 --> 00:06:50.080
Now, something looks
interesting here.

00:06:50.080 --> 00:06:55.240
We originally defined our psi,
up here, as this right here,

00:06:55.240 --> 00:06:57.090
but what is this green term?

00:06:57.090 --> 00:07:01.470
Well, what we did is we took all
of these individual terms,

00:07:01.470 --> 00:07:04.800
and these green terms here are
just taking the derivative

00:07:04.800 --> 00:07:09.400
with respect to just x on each
of these terms. Because if you

00:07:09.400 --> 00:07:12.340
take the derivative just with
respect to x of this, then the

00:07:12.340 --> 00:07:13.890
function of y is just
a constant, right?

00:07:13.890 --> 00:07:15.900
If you were to take just a
partial derivative with

00:07:15.900 --> 00:07:17.310
respect to x.

00:07:17.310 --> 00:07:19.220
So if you took the partial
derivative with respect to x

00:07:19.220 --> 00:07:22.740
of this term, you treat a
function of y as a constant.

00:07:22.740 --> 00:07:26.360
So the derivative of this would
just be f prime of x, g1

00:07:26.360 --> 00:07:29.080
of y, because g1 of y
is just a constant.

00:07:29.080 --> 00:07:30.060
And so forth and so on.

00:07:30.060 --> 00:07:33.410
All of these green terms you
can view as a partial

00:07:33.410 --> 00:07:36.430
derivative of psi with
respect to x.

00:07:36.430 --> 00:07:39.090
We just pretended like
y is a constant.

00:07:39.090 --> 00:07:43.600
And that same logic, if you
ignore this, if you just look

00:07:43.600 --> 00:07:46.890
at this part right here,
what is this?

00:07:46.890 --> 00:07:52.000
We took psi, up here, we treated
the functions of x as

00:07:52.000 --> 00:07:56.880
a constant, and we just took
the partial derivative with

00:07:56.880 --> 00:07:57.870
respect to y.

00:07:57.870 --> 00:08:00.055
And that's why the primes
are on all the g's.

00:08:00.055 --> 00:08:03.060
And then we multiply
that times dy dx.

00:08:03.060 --> 00:08:06.360
So you could write this, this
is equal to-- I'll do this

00:08:06.360 --> 00:08:12.730
green-- this green is the same
thing as the partial of psi

00:08:12.730 --> 00:08:15.830
with respect to x.

00:08:15.830 --> 00:08:21.140
Plus, what's this purple,
this part of the purple?

00:08:21.140 --> 00:08:23.850
Let me do it in a different
color, in magenta.

00:08:23.850 --> 00:08:32.409
This, right here, is the partial
of psi with respect to

00:08:32.409 --> 00:08:35.799
y, and then times dy dx.

00:08:38.890 --> 00:08:41.624
So that's essentially all I
wanted to show you right now

00:08:41.624 --> 00:08:43.150
in this video, because
I realize I'm almost

00:08:43.150 --> 00:08:43.700
running out of time.

00:08:43.700 --> 00:08:47.530
That the chain rule, with
respect to one of the

00:08:47.530 --> 00:08:53.620
variables, but the second
variable in the function is

00:08:53.620 --> 00:08:56.140
also a function of x, the
chain rule is this.

00:08:56.140 --> 00:08:59.590
If psi is a function of x and
y, and I would take not a

00:08:59.590 --> 00:09:02.380
partial derivative, I would take
the full derivative of

00:09:02.380 --> 00:09:04.620
psi with respect to x, it's
equal to the partial of psi

00:09:04.620 --> 00:09:08.830
with respect to x, plus the
partial of psi with respect to

00:09:08.830 --> 00:09:11.180
y, times dy dx.

00:09:11.180 --> 00:09:17.250
If y wasn't a function of x, or
if y if it was independent

00:09:17.250 --> 00:09:19.740
of x, than dy dx would be 0.

00:09:19.740 --> 00:09:23.220
And this term would be 0, and
then the derivative of psi

00:09:23.220 --> 00:09:26.460
with respect to x would be just
the partial of psi with

00:09:26.460 --> 00:09:27.170
respect to x.

00:09:27.170 --> 00:09:31.140
But anyway, I want you to
just keep this in mind.

00:09:31.140 --> 00:09:33.770
And in this video I didn't prove
it, but I hopefully gave

00:09:33.770 --> 00:09:35.630
you a little intuition if
I didn't confuse you.

00:09:35.630 --> 00:09:40.020
And we're going to use this
property in the next series of

00:09:40.020 --> 00:09:43.190
videos to understand exact
equations a little bit more.

00:09:43.190 --> 00:09:45.710
I realize that in this video I
just got as far as kind of

00:09:45.710 --> 00:09:46.610
giving you an intuition here.

00:09:46.610 --> 00:09:49.810
I haven't told you yet what
an exact equation is.

00:09:49.810 --> 00:09:52.400
I will see you in
the next video.


Old separable differential equations example

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:03.180
I think it's reasonable to
do one more separable

00:00:03.180 --> 00:00:07.214
differential equations problem,
so let's do it.

00:00:07.214 --> 00:00:13.430
The derivative of y with respect
to x is equal to y

00:00:13.430 --> 00:00:22.460
cosine of x divided by 1 plus
2y squared, and they give us

00:00:22.460 --> 00:00:27.080
an initial condition that
y of 0 is equal to 1.

00:00:27.080 --> 00:00:29.920
Or when x is equal to
0, y is equal to 1.

00:00:29.920 --> 00:00:32.130
And I know we did a couple
already, but another way to

00:00:32.130 --> 00:00:34.530
think about separable
differential equations is

00:00:34.530 --> 00:00:37.300
really, all you're
doing is implicit

00:00:37.300 --> 00:00:39.000
differentiation in reverse.

00:00:39.000 --> 00:00:41.930
Or another way to think about
it is whenever you took an

00:00:41.930 --> 00:00:46.340
implicit derivative, the end
product was a separable

00:00:46.340 --> 00:00:47.800
differential equation.

00:00:47.800 --> 00:00:52.130
And so, this hopefully forms a
little bit of a connection.

00:00:52.130 --> 00:00:53.030
Anyway, let's just do this.

00:00:53.030 --> 00:00:55.480
We have to separate the
y's from the x's.

00:00:55.480 --> 00:00:59.170
Let's multiply both sides
times 1 plus 2y squared.

00:00:59.170 --> 00:01:07.050
We get 1 plus 2y squared
times dy dx is equal

00:01:07.050 --> 00:01:10.360
to y cosine of x.

00:01:10.360 --> 00:01:13.130
We still haven't fully separated
the y's and the x's.

00:01:13.130 --> 00:01:17.190
Let's divide both sides of this
by y, and then let's see.

00:01:17.190 --> 00:01:23.080
We get 1 over y plus 2y squared
divided by y, that's

00:01:23.080 --> 00:01:31.020
just 2y, times dy dx is
equal to cosine of x.

00:01:31.020 --> 00:01:34.430
I can just multiply
both sides by dx.

00:01:34.430 --> 00:01:42.860
1 over y plus 2y times dy is
equal to cosine of x dx.

00:01:42.860 --> 00:01:45.380
And now we can integrate
both sides.

00:01:50.670 --> 00:01:54.090
So what's the integral of 1
over y with respect to y?

00:01:54.090 --> 00:01:57.620
I know your gut reaction is the
natural log of y, which is

00:01:57.620 --> 00:02:01.060
correct, but there's actually
a slightly broader function

00:02:01.060 --> 00:02:03.450
than that, whose derivative is
actually 1 over y, and that's

00:02:03.450 --> 00:02:06.550
the natural log of the
absolute value of y.

00:02:06.550 --> 00:02:11.640
And this is just a slightly
broader function, because it's

00:02:11.640 --> 00:02:15.550
domain includes positive and
negative numbers, it just

00:02:15.550 --> 00:02:16.240
excludes 0.

00:02:16.240 --> 00:02:18.910
While natural log of
y only includes

00:02:18.910 --> 00:02:21.160
numbers larger than 0.

00:02:21.160 --> 00:02:23.750
So natural log of absolute value
of y is nice, and it's

00:02:23.750 --> 00:02:27.200
actually true that at all
points other than 0, its

00:02:27.200 --> 00:02:28.990
derivative is 1 over y.

00:02:28.990 --> 00:02:31.140
It's just a slightly
broader function.

00:02:31.140 --> 00:02:33.550
So that's the antiderivative
of 1 over y, and we proved

00:02:33.550 --> 00:02:35.330
that, or at least we proved that
the derivative of natural

00:02:35.330 --> 00:02:38.000
log of y is 1 over y.

00:02:38.000 --> 00:02:40.890
Plus, what's the antiderivative
of 2y with

00:02:40.890 --> 00:02:41.410
respect to y?

00:02:41.410 --> 00:02:45.020
Well, it's y squared, is
equal to-- I'll do the

00:02:45.020 --> 00:02:47.230
plus c on this side.

00:02:47.230 --> 00:02:48.990
Whose derivative
is cosine of x?

00:02:48.990 --> 00:02:50.240
Well, it's sine of x.

00:02:53.290 --> 00:02:56.650
And then we could
add the plus c.

00:02:56.650 --> 00:02:59.340
We could add that
plus c there.

00:02:59.340 --> 00:03:00.870
And what was our initial
condition? y of

00:03:00.870 --> 00:03:02.000
0 is equal to 1.

00:03:02.000 --> 00:03:04.470
So when x is equal to
0, y is equal to 1.

00:03:08.175 --> 00:03:13.350
So ln of the absolute value of
1 plus 1 squared is equal to

00:03:13.350 --> 00:03:16.870
sine of 0 plus c.

00:03:16.870 --> 00:03:19.170
The natural log of one, e
to the what power is 1?

00:03:19.170 --> 00:03:24.530
Well, 0, plus 1 is-- sine of
0 is 0 --is equal to C.

00:03:24.530 --> 00:03:27.600
So we get c is equal to 1.

00:03:27.600 --> 00:03:32.710
So the solution to this
differential equation up here

00:03:32.710 --> 00:03:35.740
is, I don't even have to rewrite
it, we figured out c

00:03:35.740 --> 00:03:37.820
is equal to 1, so we can just
scratch this out, and

00:03:37.820 --> 00:03:39.020
we could put a 1.

00:03:39.020 --> 00:03:42.060
The natural log of the absolute
value of y plus y

00:03:42.060 --> 00:03:46.210
squared is equal to
sine of x plus 1.

00:03:46.210 --> 00:03:49.080
And actually, if you were to
graph this, you would see that

00:03:49.080 --> 00:03:53.430
y never actually dips below
or even hits the x-axis.

00:03:53.430 --> 00:03:55.470
So you can actually get
rid of that absolute

00:03:55.470 --> 00:03:56.700
value function there.

00:03:56.700 --> 00:03:58.150
But anyway, that's just
a little technicality.

00:03:58.150 --> 00:04:02.520
But this is the implicit form
of the solution to this

00:04:02.520 --> 00:04:03.380
differential equation.

00:04:03.380 --> 00:04:05.680
That makes sense, because the
separable differential

00:04:05.680 --> 00:04:06.800
equations are really just

00:04:06.800 --> 00:04:09.740
implicit derivatives backwards.

00:04:09.740 --> 00:04:13.120
And in general, one thing
that's kind of fun about

00:04:13.120 --> 00:04:15.810
differential equations, but
kind of not as satisfying

00:04:15.810 --> 00:04:19.490
about differential equations,
is it really is just a whole

00:04:19.490 --> 00:04:22.840
hodgepodge of tools to solve
different types of equations.

00:04:22.840 --> 00:04:27.380
There isn't just one tool or one
theory that will solve all

00:04:27.380 --> 00:04:28.180
differential equations.

00:04:28.180 --> 00:04:30.600
There are few that will solve
a certain class of

00:04:30.600 --> 00:04:32.810
differential equations, but
there's not just one

00:04:32.810 --> 00:04:34.320
consistent way to solve
all of them.

00:04:34.320 --> 00:04:37.100
And even today, there are
unsolved differential

00:04:37.100 --> 00:04:39.390
equations, where the only way
that we know how to get

00:04:39.390 --> 00:04:42.660
solutions is using a computer
numerically.

00:04:42.660 --> 00:04:44.060
And one day I'll do
videos on that.

00:04:44.060 --> 00:04:47.280
And actually, you'll find that
in most applications, that's

00:04:47.280 --> 00:04:49.460
what you end up doing anyway,
because most differential

00:04:49.460 --> 00:04:54.280
equations you encounter in
science or with any kind of

00:04:54.280 --> 00:04:56.500
science, whether it's economics,
or physics, or

00:04:56.500 --> 00:05:01.260
engineering, that they often are
unsolveable, because they

00:05:01.260 --> 00:05:04.050
might have a second or third
derivative involved, and

00:05:04.050 --> 00:05:05.090
they're going to multiply.

00:05:05.090 --> 00:05:06.760
I mean, they're just going to
be really complicated, very

00:05:06.760 --> 00:05:08.040
hard to solve analytically.

00:05:08.040 --> 00:05:10.260
And actually, you are going to
solve them numerically, which

00:05:10.260 --> 00:05:11.910
is often much easier.

00:05:11.910 --> 00:05:14.520
But anyway, hopefully at this
point you have a pretty good

00:05:14.520 --> 00:05:15.950
sense of separable equations.

00:05:15.950 --> 00:05:17.860
They're just implicit
differentiation backwards, and

00:05:17.860 --> 00:05:20.020
it's really nothing new.

00:05:20.020 --> 00:05:24.600
Our next thing we'll learn is
exact differential equations,

00:05:24.600 --> 00:05:27.210
and then we'll go off into
more and more methods.

00:05:27.210 --> 00:05:29.640
And then hopefully, by the end
of this playlist, you'll have

00:05:29.640 --> 00:05:32.470
a nice toolkit of all the
different ways to solve at

00:05:32.470 --> 00:05:34.890
least the solvable differential
equations.

00:05:34.890 --> 00:05:36.140
See you in the next video.


Old separable differential equations introduction

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:03.330
We hopefully know at this point
what a differential

00:00:03.330 --> 00:00:06.490
equation is, so now let's
try to solve some.

00:00:06.490 --> 00:00:09.000
And this first class of
differential equations I'll

00:00:09.000 --> 00:00:11.270
introduce you to, they're called
separable equations.

00:00:11.270 --> 00:00:13.570
And I think what you'll find
is that we're not learning

00:00:13.570 --> 00:00:14.420
really anything new.

00:00:14.420 --> 00:00:19.010
Using just your first year
calculus derivative and

00:00:19.010 --> 00:00:22.300
integrating skills, you can
solve a separable equation.

00:00:22.300 --> 00:00:25.240
And the reason why they're
called separable is because

00:00:25.240 --> 00:00:28.090
you can actually separate the
x and y terms, and integrate

00:00:28.090 --> 00:00:30.330
them separately to get
the solution of the

00:00:30.330 --> 00:00:31.170
differential equation.

00:00:31.170 --> 00:00:33.640
So that's separable.

00:00:33.640 --> 00:00:36.000
Separable equations.

00:00:36.000 --> 00:00:38.550
So let's do a couple, and I
think you'll get the point.

00:00:38.550 --> 00:00:42.600
These often are really more of
exercises in algebra than

00:00:42.600 --> 00:00:44.120
anything else.

00:00:44.120 --> 00:00:49.110
So the first separable
differential equation is: dy

00:00:49.110 --> 00:00:56.710
over dx is equal to x squared
over 1 minus y squared.

00:00:56.710 --> 00:00:58.420
And actually, this is a good
time to just review our

00:00:58.420 --> 00:00:59.050
terminology.

00:00:59.050 --> 00:01:00.840
So first of all, what
is the order of this

00:01:00.840 --> 00:01:02.060
differential equation?

00:01:02.060 --> 00:01:05.090
Well, the highest derivative
in it is just the first

00:01:05.090 --> 00:01:08.430
derivative, so the order
is equal to 1.

00:01:08.430 --> 00:01:11.390
So it's first order.

00:01:11.390 --> 00:01:14.020
It's ordinary, because we only
have a regular derivative, no

00:01:14.020 --> 00:01:15.530
partial derivatives here.

00:01:15.530 --> 00:01:18.130
And then, is this linear
or non-linear?

00:01:18.130 --> 00:01:20.580
Well, first you say, oh well,
you know, this looks linear.

00:01:20.580 --> 00:01:21.910
I'm not multiplying
the derivative

00:01:21.910 --> 00:01:23.050
times anything else.

00:01:23.050 --> 00:01:25.280
But if you look carefully,
something

00:01:25.280 --> 00:01:26.160
interesting is going on.

00:01:26.160 --> 00:01:28.100
First of all, you have
a y squared.

00:01:28.100 --> 00:01:29.610
And y is a dependent variable.

00:01:29.610 --> 00:01:31.440
y is a function of x.

00:01:31.440 --> 00:01:33.740
So to have the y squared, that
makes it non-linear.

00:01:33.740 --> 00:01:36.660
And even if this was a y, if you
were to actually multiply

00:01:36.660 --> 00:01:39.280
both sides of this equation
times 1 minus y, and get in

00:01:39.280 --> 00:01:42.740
the form that I showed you in
the previous equation, you

00:01:42.740 --> 00:01:47.060
would have 1 minus y squared.

00:01:47.060 --> 00:01:48.630
This is actually the first step
of what we have to do

00:01:48.630 --> 00:01:50.150
anyways, so I'll
write it down.

00:01:50.150 --> 00:01:52.020
So if I'm just multiplying both
sides of this equation

00:01:52.020 --> 00:01:56.620
times 1 minus y squared, you get
1 minus y squared times dy

00:01:56.620 --> 00:02:00.820
dx is equal to x squared.

00:02:00.820 --> 00:02:04.120
And then you immediately see
that even if this wasn't a

00:02:04.120 --> 00:02:07.020
squared here, you'd be
multiplying the y times dy dx,

00:02:07.020 --> 00:02:09.770
and that also makes it
non-linear because you're

00:02:09.770 --> 00:02:11.640
multiplying the dependent
variable times the

00:02:11.640 --> 00:02:12.690
derivative of itself.

00:02:12.690 --> 00:02:14.890
So that also makes this
a non-linear equation.

00:02:14.890 --> 00:02:17.300
But anyway, let's get back
to solving this.

00:02:17.300 --> 00:02:18.690
So this is the first step.

00:02:18.690 --> 00:02:21.480
I just multiply both sides
by 1 minus y squared.

00:02:21.480 --> 00:02:24.310
And the real end goal is just
to separate the y's and the

00:02:24.310 --> 00:02:26.200
x's, and then integrate
both sides.

00:02:26.200 --> 00:02:27.450
So I'm almost there.

00:02:27.450 --> 00:02:29.720
So now what I want to do is I
want to multiply both sides of

00:02:29.720 --> 00:02:32.940
this equation times dx, so I
have a dx here and get rid of

00:02:32.940 --> 00:02:34.375
this dx there.

00:02:34.375 --> 00:02:37.790
Let me go here, I don't want
to waste too much space.

00:02:37.790 --> 00:02:47.440
So you get 1 minus y squared dy
is equal to x squared dx.

00:02:47.440 --> 00:02:51.190
I have separated the x and
y variables and the

00:02:51.190 --> 00:02:52.420
differentials.

00:02:52.420 --> 00:02:54.510
All I did is I multiplied both
sides of this equation times

00:02:54.510 --> 00:02:56.850
dx to get here.

00:02:56.850 --> 00:03:00.160
Now, I can just integrate
both sides.

00:03:00.160 --> 00:03:01.680
So let's do that.

00:03:01.680 --> 00:03:03.080
Whatever you do to one side
of the equation you

00:03:03.080 --> 00:03:04.930
have to do the other.

00:03:04.930 --> 00:03:06.790
That's true with regular
equations or

00:03:06.790 --> 00:03:07.560
differential equations.

00:03:07.560 --> 00:03:09.180
So we're going to integrate
both sides.

00:03:09.180 --> 00:03:12.640
So what's the integral of this
expression with respect to y?

00:03:12.640 --> 00:03:13.350
Let's see.

00:03:13.350 --> 00:03:19.730
The integral of 1 is y, the
integral of y squared, well

00:03:19.730 --> 00:03:27.210
that's minus y to the
third over 3.

00:03:27.210 --> 00:03:30.230
And I'll write the plus c here
just to kind of show you

00:03:30.230 --> 00:03:32.730
something, but you really don't
have to write a plus c

00:03:32.730 --> 00:03:34.020
on both sides.

00:03:34.020 --> 00:03:37.145
I'll call the plus the
constant due to y.

00:03:37.145 --> 00:03:38.750
The y integration.

00:03:38.750 --> 00:03:40.420
You'll never see this in a
calculus class, but I just

00:03:40.420 --> 00:03:41.670
want to make a point here.

00:03:44.290 --> 00:03:45.840
I just want to show you that
our plus c has never

00:03:45.840 --> 00:03:47.720
disappeared from when we were
taking our traditional

00:03:47.720 --> 00:03:48.780
antiderivatives.

00:03:48.780 --> 00:03:50.030
And what's the derivative
of this?

00:03:50.030 --> 00:03:52.420
Well that's x to the
third over 3.

00:03:57.010 --> 00:04:01.300
And this is also going
to have a plus c

00:04:01.300 --> 00:04:03.600
due to the x variable.

00:04:03.600 --> 00:04:07.390
Now, the reason why I did this
magenta one in magenta and I

00:04:07.390 --> 00:04:09.070
labelled it like that, is
because you really just have

00:04:09.070 --> 00:04:11.380
to write a plus c on one
side of the equation.

00:04:11.380 --> 00:04:13.900
And if that doesn't make a lot
of sense, let's subtract this

00:04:13.900 --> 00:04:19.740
c from both sides, and we get y
minus-- let me scroll down a

00:04:19.740 --> 00:04:22.650
little bit, my y
looks like a g.

00:04:22.650 --> 00:04:29.370
y minus y to the third over 3
is equal to x to the third

00:04:29.370 --> 00:04:33.270
over 3 plus the constant when we
took the antiderivative of

00:04:33.270 --> 00:04:36.380
the x, minus the constant of
the antiderivative when we

00:04:36.380 --> 00:04:37.450
took the y.

00:04:37.450 --> 00:04:39.500
But these two constants,
they're just constants.

00:04:39.500 --> 00:04:40.480
I mean, we don't know
what they are.

00:04:40.480 --> 00:04:41.430
There are arbitrary constants.

00:04:41.430 --> 00:04:43.170
So we could just write
a general c here.

00:04:43.170 --> 00:04:46.170
So you could have just-- you
have to have a constant, but

00:04:46.170 --> 00:04:48.310
it doesn't have to be on both
sides of the equation, because

00:04:48.310 --> 00:04:49.580
they're arbitrary.

00:04:49.580 --> 00:04:52.140
cx minus cy, well, that's still
just another constant.

00:04:52.140 --> 00:04:53.900
And then if we want to simplify
this equation more,

00:04:53.900 --> 00:04:56.230
we can multiply both sides
of this by 3, just

00:04:56.230 --> 00:04:57.430
make it look nicer.

00:04:57.430 --> 00:05:04.510
And you get 3y minus y to the
third is equal to x to the

00:05:04.510 --> 00:05:08.580
third plus-- well, I could
write 3c here.

00:05:08.580 --> 00:05:11.190
But once again, c is an
arbitrary constant.

00:05:11.190 --> 00:05:13.570
So 3 times an arbitrary
constant, that's just another

00:05:13.570 --> 00:05:14.555
arbitrary constant.

00:05:14.555 --> 00:05:18.610
So I'll write the c there.

00:05:18.610 --> 00:05:19.690
And there you have it.

00:05:19.690 --> 00:05:22.460
We have solved this differential
equation.

00:05:22.460 --> 00:05:25.520
Although it is in implicit
form right now, and it's

00:05:25.520 --> 00:05:28.290
fairly hard to get it out
of implicit form.

00:05:28.290 --> 00:05:31.550
We could put the c on one side,
so the solution could be

00:05:31.550 --> 00:05:37.180
3y minus y to the third minus x
to the third is equal to c.

00:05:37.180 --> 00:05:38.660
Some people might like that
little bit better.

00:05:38.660 --> 00:05:39.900
But that's the solution.

00:05:39.900 --> 00:05:41.780
And notice, the solution, just
like when you take an

00:05:41.780 --> 00:05:46.930
antiderivative, the solution
is a class of implicit

00:05:46.930 --> 00:05:48.230
functions, in this case.

00:05:48.230 --> 00:05:49.320
And why is it a class?

00:05:49.320 --> 00:05:51.520
Because we have that
constant there.

00:05:51.520 --> 00:05:55.090
Depending on what number
you pick there, it

00:05:55.090 --> 00:05:56.360
will be another solution.

00:05:56.360 --> 00:05:59.440
But any constant there will
satisfy the original

00:05:59.440 --> 00:06:03.780
differential equation,
which was up here.

00:06:03.780 --> 00:06:05.540
This was the original
differential equation.

00:06:05.540 --> 00:06:08.370
And if you want to solve for
that constant, someone has to

00:06:08.370 --> 00:06:09.630
give you an initial condition.

00:06:09.630 --> 00:06:15.860
Someone has to say, well,
when x is 2, y is 3.

00:06:15.860 --> 00:06:17.920
And then you could
solve for c.

00:06:17.920 --> 00:06:19.880
Anyway, let's do another
one that gives

00:06:19.880 --> 00:06:22.950
us an initial condition.

00:06:22.950 --> 00:06:27.180
So this one's a little bit--
I'll start over.

00:06:27.180 --> 00:06:31.990
Clear image, different colors,
so I have optimal space.

00:06:31.990 --> 00:06:36.690
So this one is the first
derivative of y with respect

00:06:36.690 --> 00:06:49.940
to x is equal to 3x squared
plus 4x plus 2 over 2

00:06:49.940 --> 00:06:52.100
times y minus 1.

00:06:52.100 --> 00:06:54.030
This is a parentheses, not
an absolute value.

00:06:54.030 --> 00:06:56.280
And they give us initial
conditions.

00:06:56.280 --> 00:07:00.640
They say that y of 0 is
equal to negative 1.

00:07:00.640 --> 00:07:03.010
So once we solve this
differential equation, and

00:07:03.010 --> 00:07:05.860
this is a separable differential
equation, then we

00:07:05.860 --> 00:07:08.830
can use this initial condition,
when x is 0, y is

00:07:08.830 --> 00:07:10.520
1, to figure out the constant.

00:07:10.520 --> 00:07:12.220
So let's first separate
this equation.

00:07:12.220 --> 00:07:15.610
So let's multiply both sides
by 2 times y minus 1.

00:07:15.610 --> 00:07:25.340
And you get 2 times y minus 1
times dy dx is equal to 3x

00:07:25.340 --> 00:07:28.620
squared plus 4x plus 2.

00:07:28.620 --> 00:07:30.040
Multiply both sides times dx.

00:07:30.040 --> 00:07:33.560
This is really just an
exercise in algebra.

00:07:33.560 --> 00:07:39.030
And I can multiply this one out,
too, you get 2y minus 2,

00:07:39.030 --> 00:07:40.940
that's just this, dy.

00:07:40.940 --> 00:07:47.390
I multiplied both sides times
dx, so that equals 3x squared

00:07:47.390 --> 00:07:52.730
plus 4x plus 2 dx.

00:07:52.730 --> 00:07:54.740
I have separated
the equations.

00:07:54.740 --> 00:07:57.990
I've separated the independent
from the dependent variable,

00:07:57.990 --> 00:08:00.690
and their relative
differentials, and so now I

00:08:00.690 --> 00:08:01.570
can integrate.

00:08:01.570 --> 00:08:02.840
And I can integrate
in magenta.

00:08:06.500 --> 00:08:08.950
What's the antiderivative
of this expression

00:08:08.950 --> 00:08:09.650
with respect to y?

00:08:09.650 --> 00:08:10.080
Well, let's just see.

00:08:10.080 --> 00:08:14.100
It's y squared minus 2y.

00:08:14.100 --> 00:08:15.450
I won't write the plus
c, I'll just do it on

00:08:15.450 --> 00:08:17.200
the right hand side.

00:08:17.200 --> 00:08:19.995
That is equal to 3x squared.

00:08:19.995 --> 00:08:25.400
Well, the antiderivative is x to
the third, plus 2x squared,

00:08:25.400 --> 00:08:29.580
plus 2x plus c.

00:08:29.580 --> 00:08:32.010
And that c kind of takes care of
the constant for both sides

00:08:32.010 --> 00:08:34.159
of the equation, and hopefully
you understand why from the

00:08:34.159 --> 00:08:35.080
last example.

00:08:35.080 --> 00:08:39.270
But we can solve for c using the
initial condition y of 0

00:08:39.270 --> 00:08:41.480
is equal to negative 1.

00:08:41.480 --> 00:08:42.870
So let's see.

00:08:42.870 --> 00:08:44.950
When x is 0, y is negative 1.

00:08:44.950 --> 00:08:50.090
So let's put y as negative 1,
so we get negative 1 squared

00:08:50.090 --> 00:08:55.870
minus 2 times negative 1, that's
the value of y, is

00:08:55.870 --> 00:08:57.500
equal to when x is equal to 0.

00:08:57.500 --> 00:09:00.890
So when x is equal to 0, that's
0 to the third plus 2

00:09:00.890 --> 00:09:04.760
times 0 squared plus
2 times 0 plus c.

00:09:04.760 --> 00:09:06.260
So this is fairly
straightforward.

00:09:06.260 --> 00:09:09.280
All of these, this is all 0.

00:09:09.280 --> 00:09:14.940
This is, let's see, negative
1 squared, that's 1.

00:09:14.940 --> 00:09:20.600
Minus 2 times minus 1, that's
plus 2, is equal to c.

00:09:20.600 --> 00:09:25.170
And we get c is equal to 3.

00:09:25.170 --> 00:09:28.710
So, the implicit exact solution,
the solution of our

00:09:28.710 --> 00:09:30.080
differential equation-- remember
now, it's not a

00:09:30.080 --> 00:09:35.880
class, because they gave us an
initial condition-- is y

00:09:35.880 --> 00:09:43.170
squared minus 2y is equal to x
to the third plus 2x squared

00:09:43.170 --> 00:09:46.260
plus 2x plus 3.

00:09:46.260 --> 00:09:48.730
We figured out that's
what c was.

00:09:48.730 --> 00:09:50.460
And actually, if you want, you
could write this in an

00:09:50.460 --> 00:09:52.800
explicit form by completing
the square.

00:09:52.800 --> 00:09:54.180
This is just algebra
this point.

00:09:54.180 --> 00:09:54.510
You're done.

00:09:54.510 --> 00:09:55.450
This is an implicit form.

00:09:55.450 --> 00:09:58.450
If you wanted to make it
explicit, you could add 1 to

00:09:58.450 --> 00:09:58.890
both sides.

00:09:58.890 --> 00:10:00.660
I'm just completing
the square here.

00:10:00.660 --> 00:10:04.480
So y squared minus 2y plus 1.

00:10:04.480 --> 00:10:07.060
If I add 1 to that side, I have
to add 1 to this side, so

00:10:07.060 --> 00:10:13.090
it becomes x to the third plus
2x squared plus 2x plus 4.

00:10:13.090 --> 00:10:14.940
I just added 1 to both sides
of the equation.

00:10:14.940 --> 00:10:15.770
Why did I do that?

00:10:15.770 --> 00:10:17.680
Because I wanted this
side to be a perfect

00:10:17.680 --> 00:10:19.240
square in terms of y.

00:10:19.240 --> 00:10:24.590
Then I can rewrite this side as
y minus 1 squared is equal

00:10:24.590 --> 00:10:30.500
to x to the third plus 2x
squared plus 2x plus 4.

00:10:30.500 --> 00:10:35.110
Then I could say y minus 1 is
equal to the plus or minus

00:10:35.110 --> 00:10:39.950
square root of x to the
third plus 2x squared

00:10:39.950 --> 00:10:43.210
plus 2x plus 4.

00:10:43.210 --> 00:10:46.520
I can add 1 to both sides, and
then I can get y is equal to 1

00:10:46.520 --> 00:10:52.080
plus or minus the square root
of x to the third plus 2x

00:10:52.080 --> 00:10:55.926
squared plus 2x plus 4.

00:10:55.926 --> 00:10:58.470
And it has plus or minus here,
and if we have to pick one of

00:10:58.470 --> 00:11:00.335
the two, we'd go back to
the initial condition.

00:11:02.860 --> 00:11:08.800
Well, our initial condition told
us that y of 0 is equal

00:11:08.800 --> 00:11:10.200
to negative 1.

00:11:10.200 --> 00:11:16.710
So if we put 0 here for x, we
get y is equal to 1 plus or

00:11:16.710 --> 00:11:19.065
minus 0 plus 4.

00:11:19.065 --> 00:11:22.100
So 1 plus or minus 4.

00:11:22.100 --> 00:11:24.740
So if y is going to be equal to
negative 1, so we get y is

00:11:24.740 --> 00:11:28.580
equal to 1 plus or
minus-- sorry, 2.

00:11:28.580 --> 00:11:32.090
If this is going to be equal to
negative 1, then this has

00:11:32.090 --> 00:11:33.830
to be 1 minus 2.

00:11:33.830 --> 00:11:37.160
So the explicit form that
satisfies our initial

00:11:37.160 --> 00:11:39.490
condition, and we're getting a
little geeky here, you could

00:11:39.490 --> 00:11:42.970
get rid of the plus, it's 1
minus this whole thing.

00:11:42.970 --> 00:11:45.340
That's what satisfies our
initial condition.

00:11:45.340 --> 00:11:49.430
And you could figure out where
it's satisfied, over what

00:11:49.430 --> 00:11:50.910
domain is it satisfied.

00:11:50.910 --> 00:11:53.300
Well, that's satisfied when this
term is positive, this

00:11:53.300 --> 00:11:56.130
becomes negative, and you get
it's undefined in reals, and

00:11:56.130 --> 00:11:56.470
all of that.

00:11:56.470 --> 00:11:57.950
But anyway, I've run
out of time.

00:11:57.950 --> 00:11:59.970
See you in the next video.