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Here is this equation

which is sadly, not exact, because

Our job is to use some magic and make it exact.

Let’s try multiplying the equation by x.

Let’s see whether this multiplication by x was good for the equation.

It seems that it was.

This is now an exact equation, and its solution is:

At the end we find out what is.

Well, it seems it was effective to multiply by x.

This is lovely, but begs the question: why we chose x for the multiplication.

If the equation is not exact, then we try to make it exact by using an integrating factor.

To find the integrating factor, first we compute these:

There is nothing to worry about, everything will be all right soon.

If the first one of these contains only y,

or the second one contains only x,

then there is hope for finding the integrating factor.

Now the first one contains both x and y, so that is not useful for us.

But the second one is good.

And the integrating factor...

Finding the integrating factor

Here comes another equation.

Let’s check whether it is exact.

Well, not really.

So, here comes the integrating factor.

The first one should only contain x...

so, sadly, this is not good.

The second one is promising...

Now, this equation is exact.

So, let’s get down to solving it:

Bad news. This is partial integration.

And there is this as well.

Well, it seems , therefore is just some constant.

Here is an equation.

Let’s check whether it is exact.

It seems it is not exact.

No problem, here comes the integrating factor.

After some transformation...

Now this equation is exact.

So, let’s get down to solving it:

Finally, let’s differentiate this with respect to y, to see what is going on with .

And then we have this equation:

Let's see whether it is exact.

Well, no.

No problem, here comes the integrating factor.

Here, it doesn’t matter which one we use.

But this one looks easier.

Now we are ready for the solution.

 

The integrating factor

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