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Say we have

We will rewrite it to

or

This is usually the form it is expressed in. Ok. How to solve this? To figure that out we will work a bit backwards. We will somewhat do a trick similar to completing the square. Let us look at a special case of the product rule

This is kind of almost what we have on the LHS of our equation. Would it not be nifty if we could make it have that form? Then we could collapse it to a single derivative, and then we could take the antiderivative of both sides to find our solution. What is missing is some factor in front of *y*´. So let’s multiply through with some factor *I*. We get

So, to be able to collapse it to what we want to have we must have that

Combining the two equations above we get

or

We can solve this to get

or

where, as usual, we let the constant absorb the sign.

Back to our equation, multiplying through be the above we get

As we can see we can immediately divide away the constant, so we could instead have used

to get

This could now be collapsed to

I.e. the first term without the derivative. Integrating (taking the antiderivative) of both sides gives us

or

or, written out with all our integrals in full glory

Quite a formidable formula that you thankfully normally are not supposed to remember. Instead you remember that you have to multiply through by

that we will call the integrating factor.

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