0:22

I will check the theorem through the examples, right, okay.

First let's try to solve the following differential equation,

right, it looks like it's a complicated one.

y prime is equal to e to the -y-

y sine xy, over xe to the -y

+ x sine xy + 2y, okay?

y to this to the differential equation in differential form, okay?

So you have, okay, (e to the -y-

y sin xy) dx- (xe to the y + x sin

xy + 2y) dy = 0, right, okay?

1:43

What is the impartial y, okay, dM/dy.

Can you take a differentiation of this expression with respect to y,

dM over dY is equal to from this is -e to the -y-

sine xy- y times derivative of sine xy with respect to y.

It's equal to x times the cosine of xy, so

that dM over dy will be equal to this one, okay?

On the other hand, what is N,

N is equal to negative this expression, right, okay?

Take an extra partial derivative of that, so dN over dx,

from this one, you get negative exponential negative y,

from this, negative sine xy.

2:41

-x times derivative of sine xy with respect to x is y times

the cosine of xy, so you have this term down there, okay?

And the extra part of derivative of 2y that is equal to 0.

So you can confirm it, dN over dx is again this quantity, right?

In other words, dM over dy and dN over dx,

they coincide on the whole plane, right, that means what?

The given differential equation is exact, right, in the plane, okay.

Then how to solve this differential equation, okay,

the problem is reduced to how to find the capital F.

Whose total differential is this right hand side, okay?

3:38

But, What do you mean by, this differential equation exact?

This differential equation is exact, it means there's a certain function F, okay.

We just checked that the given differential

equation is exact, so that there is certain

function F satisfying dF over dx = M.

M = e to the -y- y sine xy, and

dF over dy, that is equal to N,

N is equal to negative xe to

the the -y + x sine xy + 2y,

okay, we get this one.

4:31

As we have done already in the proof of the theorem, okay,

how to solve this step, okay.

Starting from this equation right, starting from the first equation,

what is F is equal to?

This is the e to the -y- y sine xy,

and the integration respect to dx.

Plus you need some integral constant which can be arbitrary function of g, right, or

more precisely, arbitrary differential function of y, okay?

5:06

So that is equal to, first let's take the integration here, right, from this one.

Because we are taking integration with respect to the x variable,

this is e to the -y times x, right.

And -y, what's the antiderivative of sine xy with respect to dy, then this is what?

The negative cosine of xy, then the 1 over y, okay, you get this one.

Okay, are you following me, right, so plus g(y), okay?

Plus g(y), so this negative and another negative down there makes a plus, right.

And this y and the y in the bottom, they canceled out.

So you get, finally,

x e to the -y + cosine xy + g(y), okay?

That is a candidate for F, right, that is the candidate for F.

But this F must also satisfy another condition,

say dF over dy is equal to this one.

Okay, so plug in that expression,

F is equal to this one, into this second equation.

Then you have, from this expression, dF over dy, okay let's compute it.

From this dF over dy, from here, you get -xy times e to the -y.

And therefore with g(y) down there, derivative of cosine is -sine xy,

and divided chain rule, derivative of this side, that is x.

So you get this expression, and the plus derivative of this with respect to y,

that is essentially g'(y), okay?

Then must have equal to the same thing as down there, okay?

What is this one, that is equal to -x

e to the -y -x sine xy and -2 over y, okay?

Unfortunately, I made some mistake somewhere,

because I cannot have this y down there.

Where did I such a mistake, that is just x e to the -y, right, okay.

x e to the -y, where does this y come from?

So where did I make a mistake right from here, dF over dy.

That is x times -1 times e to the -y, right, I should not have the y down there,

that was my mistake, so let me erase this one.

8:04

-x times e to the -y, and -x times e to the -y, they canceled out.

And again, -x times sine xy and

-x times sine xy, they canceled out again.

So what does that mean, then, okay, finally we get simply,

therefore, g'(y) = -2y, okay?

What does that mean, g(y) = -y squared, for example, okay?

I'm not choosing anything,

I'm choosing the integral constant to be equal to 0, okay?

It doesn't matter, because finally, with this choice of g, okay,

8:50

my choice of the capital F is, right?

Now finally, the F(x, y),

my F is equal to, right, down here,

this is x times e to the -y + cosine (xy),

okay, and the g(y) = -y squared.

So this is the quantity for F, and

finally our general solution will be F(x,y) = c,

where c is obviously constant, right, so let me write it here, right.

This is the general solution to the problem given down there,

y' is equal to that, and this is the general solution, right?

Including one arbitrary constant,

this is the general solution to the given differential equation, right, okay?