Power series differential equations question

[JamesGoh]

Homework Statement

Using power series, find solutions to the following DE

y' + y= x^2, y(1)= 2 and xo=1

Homework Equations

y(x)=an$\sum$(x-xo)^n for n=1 to infinity

The Attempt at a Solution

See the attachment

NOTE: I only want to find a way to collect all the x terms in the series as one like group. Similary, I want to do the same with the coefficeints

I will try to solve the rest myself

 

[CompuChip]

I think the usual way to do this is:
$$y'(x) + y(x) = \sum_{n = 0}^\infty a_n n (x - x_0)^{n - 1} + \sum_{n = 0}^\infty a_n (x - x_0)^{n}$$
as you write. Now note that for n = 0, the first term of the first sum vanishes, so you can rewrite this to
$$\sum_{n = 0}^\infty a_n n (x - x_0)^{n - 1} = \sum_{n = 1}^\infty a_n n (x - x_0)^{n - 1} = \sum_{n = 0}^\infty a_{n + 1} (n + 1) (x - x_0)^{n}.$$
Now you can merge the sums again:
$$y'(x) + y(x) = \sum_{n = 0}^\infty b_n (x - x_0)^n$$
and derive a recursive equation for the $a_n$.

 

[JamesGoh]

thanks