For “fun” over the last few days, I’ve been looking at some of my old undergraduate textbooks and trying to see what I remembered. One thing I was vaguely still aware of (from 1996 or so) was the use of integrating factors to solve simple first order ODEs. So, once I figured them out again, I thought I’d write it all down here to help me remember for next time.

So, imagine you have an ODE like this:

Now, it would be nice to simplify the left hand side – to fo this, consider the following:

which is the left hand side, multiplied by the integrating factor . So, if we multiply our original ODE by the integrating factor, we can then make the simplification:

where c is the arbitrary constant from the integration of the right hand side, and can often be evaluated with initial (or boundary) conditions.

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