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!
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.
If you must use a power series then write [tex]y= \sum_{n= 0}^\infty a_nx^n[/tex] so that [tex]y'= \sum_{n= 1}^\infty na_nx^{n-1}[/tex] Write the right hand side as a power series in x (in your example, [itex]-x^2+ 2/x+ 3[/itex], 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 [itex]a_0[/itex]- that's your constant of integration.