- #1

- 57

- 8

## Homework Statement

Solve y''+(cosx)y=0 with power series (centered at 0)

## Homework Equations

y(x) = Σ a

_{n}x

^{n}

## The Attempt at a Solution

I would just like for someone to check my work:

I first computed (cosx)y like this:

(cosx)y = (1-x

^{2}/2!+x

^{4}/4!+ ...)*(a

_{0}+a

_{1}x+a

_{2}x

^{2}+...)

=a

_{0}+a

_{1}x+(a

_{2}-a

_{0}/2)x

^{2}+(a

_{3}-a

_{1}/2)x

^{3}+...

Then I computed y'' as follows:

y'' = 2a

_{2}+6a

_{3}x+12a

_{4}x

^{2}+...

I assembled everything (y''+ycosx = 0):

(a

_{0}+2a

_{2})+(6a

_{3}+a

_{1})x+(12a

_{4}+a

_{2}-a

_{0}/2)x

^{2}+... = 0

Finally I computed the constants:

a

_{0}+2a

_{2}=0 => a

_{2}=-a

_{0}/2

6a

_{3}+a

_{1}=0 => a

_{3}=-a

_{1}/6

12a

_{4}+a

_{2}-a

_{0}/2)x

^{2}=0 => a

_{4}=a

_{0}/12

...

and so on for a couple of more terms.

My final answer is then:

y(x) = a

_{0}(1-x

^{2}/2+x

^{4}/12-x

^{6}/80+...)+a

_{1}(x-x

^{3}/6+x

^{5}/30+...)

Does this make sense? Is there a 'less-messy' way of doing this?