Power series method and various techniques

[Jen_Jer_888]

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!

 

[hunt_mat]

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.

 

[HallsofIvy]

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.