# Power series method and various techniques

 Math Emeritus Sci Advisor Thanks PF Gold P: 39,568 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.