Contents of this Calculus 2 episode:

Exact equations, Finding F(x,y), Integration, Solution, Doubble integrals, Integrate with respect to x, Integrate with respect to y.

Exact differential equations

This equation

is exact, if there exists a function such that

Then, the solution for the equation is that function:

So, solving an exact differential equation means finding this function.

However, first we should test the equation to see whether it is exact.

We can do it in two ways.

One is differentiation, the other is integration.

Well, this will be a lot more understandable once we look at the family tree of the participants.

Functions and in the equation should have a common ancestor, namely .

We can check that by integration.

At the same time, they also should have a common descendant:

(What a horrible incest!!!)

We can check that by differentiation.

Well, differentiation is easier.

So, first differentiate and , to see if the equation is exact indeed,

and then integrate them to get the solution.

That’s a brilliant plan, so let’s solve a problem.

Here is an equation:

Let's see whether it is exact.

This equation is exact, if

Well, it seems yes.

The solution of exact equations is where

We obtain the solution by integration:

Integrate with respect to x,

in this case y behaves like a constant.

But because y behaves just like a constant, it is possible that this is not simply , but it contains y as well.

We can verify this by differentiating this with respect to y, and see what we get as a result.

Well, theoretically we should get .

Let’s compare to the original.

It seems that

Well, this seems to be done.

We could try writing the solution in explicit form, too,

namely, in a form where y is expressed.

Sadly, this cannot always be done.

But now it does work.

Let's see another one, too.

Let’s check whether it is exact.

It seems to be exact, so let’s get down to solving it.

Integrate with respect to x,

in this case y behaves like a constant.

And here comes this , where it is possible that this is not simply a , but it contains y as well.

Let's see what it is this time.

Here, y cannot be expressed, so we cannot write the solution in explicit form.

Finally, let's see one more equation.

First check whether the equation is exact.

Sadly, these are not equal, so the equation is not exact.

Let’ see what can be done in such cases.

The next slides will discuss that.

Exact equations, Finding F(x,y), Integration, Solution, Doubble integrals, Integrate with respect to x, Integrate with respect to y.

Calculus 2 episode