Understanding Power Series Solutions for Differential Equations

[Fuzedmind]

Homework Statement

Solve this equation using power series: y'' + y = x

Homework Equations

none

The Attempt at a Solution

I am confused about the x on the RHS of the equation. If the equation was y'' + y = 0, I would have no problem solving it. I am just a little confused about how the x fits into the equation for c_n.

if the equation were y'' + y = 0, then c_{n + 2} = c_n/(n+1)(n+2). How does the x fit into this?

 

[Mark44]

You should be able to split up the series you get into three parts. One of those parts will be very short, with just a single term.

Show us what you have for your series.

 

[Fuzedmind]

I'm a newb and don't know how to use this website, so here goes:

SUM[ (n+2)(n+1)c_n+2xⁿ] + SUM[c_nxⁿ] = x

 

[Fuzedmind]

$$\ \ \sum_{n=0}^{\infty}{(n+2)(n+1)c_{n+2}x^n} \ \ + \ \ \sum_{n=0}^{\infty}{c_{n}x^n} \ \ = x$$

Here it is in the non-retarded version

 

[Mark44]

Try expanding your two series and then grouping the results by powers of x. Be sure to take the x on the right side into account.

 

[Fuzedmind]

I kind of understand what you're saying, but could you try being a little more specific? I am terrible with series.