## Power series method and various techniques

I know how to do problems like y' + y = 0 where you can replace y' and y with a series in sigma notation, manipulate and compare coefficients.

But how do you solve a differential by power series that does not also include y or a higher order derivative? For example, y' = -(x^2) + 2/x + 3. What power series techniques can be employed here?

Any help would be appreciated!
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 Recognitions: Homework Help You not normally use a power series solution for first order differential equations, they're normally for second order and above. In your example you can integrate straight away to find your solution.
 Recognitions: Gold Member Science Advisor Staff Emeritus If you must use a power series then write $$y= \sum_{n= 0}^\infty a_nx^n$$ so that $$y'= \sum_{n= 1}^\infty na_nx^{n-1}$$ Write the right hand side as a power series in x (in your example, $-x^2+ 2/x+ 3$, write 2/x as a power series using the generalized binomial theorem) and compare coefficients of the same power. The only difference is that now, you will have a single equation for each "n" rather than a recursion relation. Of course, there will be no equation involving $a_0$- that's your constant of integration.

 Tags power series, power series method, series, series solution

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