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Pseudo Statistic

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I was reading this math book once... and it had a method for solving differential equations of 1st (And maybe 2nd? I don't remember) order by using simple Taylor series...

I didn't even have to understand much of what was going on, except that I followed some simple rule and I ended up with an accurate solution.

Anyone know where I can find more information about this method? All of the other ones I'm finding show power series which look fairly tedious to compute.

Thanks.