Solve y''+(cosx)y=0 with power series (centered at 0)
y(x) = Σ anxn
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-x2/2!+x4/4!+ ...)*(a0+a1x+a2x2 +...)
Then I computed y'' as follows:
y'' = 2a2+6a3x+12a4x2+...
I assembled everything (y''+ycosx = 0):
(a0+2a2)+(6a3+a1)x+(12a4+a2-a0/2)x2+... = 0
Finally I computed the constants:
a0+2a2=0 => a2=-a0/2
6a3+a1=0 => a3=-a1/6
12a4+a2-a0/2)x2=0 => a4=a0/12
and so on for a couple of more terms.
My final answer is then:
y(x) = a0(1-x2/2+x4/12-x6/80+...)+a1(x-x3/6+x5/30+...)
Does this make sense? Is there a 'less-messy' way of doing this?