Differential Equations and Power Series

[kehler]

Homework Statement

Solve the differential equation f' = 2xf² with the initial condition f(0)=1 in the following way:
i) First, assume that there is a solution given by a power series
f(x) =

with a positive radius of convergence. SUbstitude this into the differential equation and figure out the coefficients a_n. (it is enough to guess a pattern - you do not have to prove that your guess is correct)

The Attempt at a Solution

I know f' = sigma(from n=1 to infinity)na_nx^n-1.
So sigma(from n=1 to infinity)na_nx^n-1 = 2x

^2
I substituded x=0 into f(x) and found that a₀=1
I don't really know where to go from here :S. How do I figure out the coefficients??

 

[Dick]

Let's skip the infinite series notation, it will just get in the way here. This problem is tough to do this way because of the f^2. You have f=1+a1*x+a2*x^2+... so f'=a1+2*a2*x+... That's equal to 2*x*(1+a1*x+a2*x^2+...)^2. You want to equate equal powers of x on either side. So you have to square out that expression to get the coefficients of the power x^k up to whatever k you feel you need to solve for.

 

[kehler]

Thanks! That really helped :). Hm the pattern I am getting is 1, 0, 1, 0, 1... What function gives you alternating 0 and 1's?? :S
Would cos²(n pi/2) work?

 

[Dick]

I'm getting 1,0,1,0,1... too. It's not a cos, it's a geometric series. Add it up or try solving the original differential equation by separation of variables. Either way, you find it's non-series form is 1/(1-x^2).