Hovedindhold

## Eksakte ligninger og integrerende faktorer

Aktuel tid:0:00Samlet varighed:12:09

# Eksakte ligninger eksempel 1

## Video udskrift

OK, I filled your brain with a
bunch of partial derivatives and psi's, with respect
to x's and y's. I think now it's time to
actually do it with a real differential equation,
and make things a little bit more concrete. So let's say I have the
differential, y, the differential equation, y cosine
of x, plus 2xe to the y, plus sine of x, plus-- I'm
already running out of space-- x squared, e to the y,
minus 1, times y prime, is equal to 0. Well, your brain is already,
hopefully, in exact differential equations mode. But if you were to see this
pattern in general, where you see a function of x and y,
here-- this is just some function of x and y-- and then
you have another function of x and y, times y prime, or times
dy, d of x, your brain should immediately say if this
is inseparable. And I'm not going to try to
make it separable, just because that'll take
a lot of time. But if it's not separable, your
brain said, oh, maybe this is an exact equation. And, you say, let
me test whether this is an exact equation. So if this is an exact equation,
this is our function M, which is a function
of x and y. And this is our function
N, which is a function of x and y. Now, the test is to see if the
partial of this, with respect to y, is equal to the partial
of this, with respect to x. So let's see. The partial of M, with respect
to y, is equal to-- let's see, y is-- so this cosine of x is
just a constant, so it's just cosine of x. Cosine of x plus-- now,
what's the derivative? Well, 2x is just a constant,
what's the derivative of e to the y, with respect to y? Well, it's just e
to the y, right? So we have the constant on
the outside, 2x times the derivative, with respect to
y, so it's 2xe to the y. Fair enough. Now, what is the partial
derivative of this, with respect to x? So N sub x, or the partial of
N, with respect to x-- so what's the derivative of sine
of x, with respect to x? Well, that's easy, that's cosine
of x, plus 2x times e to the y, right? e and y is just
a constant, because y is constant when we're taking the
partial, with respect to x. So plus 2xe to the y. And then minus 1, the derivative
of a constant, with respect to anything
is going to be 0. So the derivative of N-- the
partial of N, with respect to x, is cosine of x, plus 2xe to
the y, which, lo and behold, is the same thing as the
derivative, the partial of M, with respect to y. So there we have it. We've shown that M of y is equal
to-- or the partial of M, with respect to y-- is equal
to the partial of N, with respect to x, which
tells us that this is an exact equation. Now, given that this is an exact
equation-- oh, my wife snuck up behind me,
I was wondering. I thought there was
some critter in my house, or something. Anyway, so we know that this is
an exact equation, so what does that tell us? Well, that tells us that there's
some psi, where the partial derivative of psi, with
respect to x, is equal to M, and the partial derivative of
psi, with respect to y, is equal to N. And if we know that psi,
then we can rewrite our differential equation as the
derivative of psi, with respect to x, is equal to 0. So let's solve for psi. So we know that the partial of
psi, with respect to x, is equal to M. So we could write that. We could write the partial of
psi, with respect to x, is equal to M, which is y cosine
of x, plus 2xe to the y. That's just here. That's my M of x. We could have done
it the other way. We could have said the partial
of y-- the partial of psi, with respect to y, is this
thing over here. But let's just do it with x. Now, to at least get kind of a
first approximation of what size-- not an approximation, but
to start to get a sense of it-- let's take the derivative
of both sides, with respect to-- sorry, take the
antiderivative-- take the integral of both sides,
with respect to x. So if you take the derivative of
this, with respect to x, if you integrate-- sorry, if
you were to take the antiderivative of this,
with respect to x. So let me just write
that down. The partial, with
respect to x. We're going to [? integrate ?] it, with respect to x. That is going to be equal to
the integral of this whole thing, with respect to x. Cosine of x plus 2xe to the y. We're integrating with
respect to x. And normally when you integrate
with respect to x, you'd say, OK, plus c, right? But it actually could be a
plus-- since this is a partial, with respect to x, we
could have had some function of y here in general, because
y, we treat it as a constant, right? And that makes sense, because
if you were to take the partial of both sides of this,
with respect to x, if you were to take the partial of a
function that is only a function of y, with respect
to x, you would have gotten a 0 here. So when you take the
antiderivative, you were, like, oh well, there might have
been some function of y here that we lost when
we took the partial, with respect to x. So anyway, this will
simplify to psi. psi is going to be equal to the
integral, with respect to x, or the antiderivative, with
respect to x, here, plus some function of y that we might
have lost when we took the partial, with respect to x. So let's do that. Let's figure out
this integral. I'll do it in blue So y
is just a constant. So the antiderivative of y
cosine of x, is just y sine of x, plus-- either the y
is constant, so 2x. The antiderivative of 2x, with
respect to x, is x squared, so it's x squared e to the y. And then plus some
function of y. And if you want to verify this,
you should take the partial of this, with
respect to x. If you take the partial of
this, with respect to x, you're going to get this in
here, which is our function, M, up here. And then when you take the
partial of this, with respect to x, you'll get 0, and
it'll get lost. OK, so we're almost there. We've almost figured out our
psi, but we still need to figure out this function of y. Well, we know that if we take
the partial of this, with respect to y, since this
is an exact equation, we should get this. We should get our N function. So let's do that. So the partial-- I'll switch
notation, just to expose you to it-- the partial psi, with
respect to y, is going to be equal to-- so here, y sine
of x, sine of x is just a constant. y is just y, so the
derivative of this, with respect to y, is
just sine of x. Plus the derivative of e to the
y is e to the y. x squared is just a constant. So it's just x squared e to
the y, plus-- what's the partial of f of y, with
respect to y? It's going to be f prime of y. Well, what did we do? We took M, we integrated with
respect to x, and we said, well, we might have lost some
function of y, so we added that to it. And then we took the partial of
that side that we've almost constructed, and we took
the partial of that, with respect to y. Now, we know, since this is
exact, that that is going to equal our N. So our N is up there. Cosine of x plus-- So that's
going to be equal to-- I want to make sure I can read it up
there-- to our N, right? Oh no, sorry. N is up here. Our N is up here. Sine of x-- let me write
that-- sine of x plus x squared, e to the y, minus 1. So sine of x plus x squared,
e to the y, minus 1. That was just our N,
from our original differential equation. And now we can solve
for f prime of y. So let's see, we get sine of x
plus x squared, e to the y, plus f prime of y, is equal to
sine of x plus x squared, e to the y, minus 1. So let's see, we can delete
sine of x from both sides. We can delete x squared e to
the y from both sides. And then what are
we left with? We're left with f prime
of y is equal to 1. And then we're left with f of
y is equal to-- well, it equals y plus some constant,
c, right? So what is our psi now? We wrote our psi up here, and we
had this f of y here, so we can rewrite it now. So psi is a function of x and
y-- we're actually pretty much almost done solving it-- psi
is a function of x and y is equal to y sine of x, plus x
squared, e to the y, plus y-- oh, sorry, this is f prime
of y, minus 1. So this is a minus 1. So this is a minus y plus c. So this is going to be
a minus y plus c. So we've solved for psi. And so what does that tell us? Well, we said that original
differential equation, up here, using the partial
derivative chain rule, that original differential equation,
can be rewritten now as the derivative dx of psi is
equal to-- psi is a function of x and y-- is equal to 0. Or if you were to integrate both
sides of this, you would get that psi of xy is equal
to c is a solution of that differential equation. So if we were to set this is
equal to c, that's the differential equation. So we could say, y sine of x
plus x squared, e to the y, minus y-- now we could say, plus
this c-- plus this c, you call it c1, is equal to c2. Well, you could subtract the c's
from both sides, and just be left with a c at the end. But anyway, we have solved
this exact equation, one, first, by recognizing it was
exact, by taking the partial of this, with respect to y, and
seeing if that was equal to the partial of N,
with respect to x. Once we saw that they were
equal, we're like, OK, this is going to be exact. So let's figure out psi. Since this is exact, M is going
to be the partial of psi, with respect to x. N is the partial of psi,
with respect to y. Then to figure out y, we
integrated M, with respect to x, and we got this. But since we said, oh, well,
instead of a plus c, it could have been a function of y there,
because we took the partial, with respect to x, so
this might have been lost. To figure out the function of y,
we then took our psi that we figured out, took the partial
of that, with respect to y, got this. And we said, this was an exact
equation, so this is going to equal our N of x y. We set those equal to each
other, and then we solved for f of y. And then we had our final psi. Our final psi was this. And then the differential
equation, because of the chain rule of partial derivatives,
we could rewrite the differential equation as this. The solution is this, and so
this is the solution to our differential equation. See you in the next video.