# Homework Help: Series Solution Near an Ordinary Point

1. Apr 12, 2012

### Ozymandius

1. The problem statement, all variables and given/known data
Determine φ''(x0), φ'''(x0), and φ(4)(x0) for the given point x0 if y=φ(x) is a solution of the given initial value problem.

y'' + (sinx)y' + (cosx)y = 0 y(0) = 0; y'(0) = 1

2. Relevant equations
y = φ(x) = Ʃan(x-x0)n

3. The attempt at a solution
I started off by differentiating y to get y' and y''. I then plugged those into the original equation, adjusted the numbers a bit to get the entire equation to fit into one summation. From there, I factored out xn, then set the bulk of the equation equal to 0. From there, I solved for an+2, plugged in n=0 to get:

a2 = (-(sinx)a1 - (cosx)a0))/2

This is where I get stuck. I am unsure as to what I'm supposed to do from here, although I have ideas. Am I just supposed to somehow solve for a2 and plug it into the φ''(x) equation? If so, where do I get a1 and a0 from?

Also, I apologize for not knowing how to do the graphic code to make it look as it looks on paper. Thank you!

UPDATE: Assuming I am correct, I have solved for a2, a3, and a4 when setting a0 = y(0) and a1 = y'(0).
How do I use these newfound numbers to solve for the solutions?

a2 = 0
a3 = -1/6
a4 = 0

Last edited: Apr 12, 2012