# Power series for ODE

1. May 15, 2010

### filter54321

1. The problem statement, all variables and given/known data
Find the first 6 terms of the power series expansion centered at 0 for the general solution for y
-xy'=0. Then find the coefficient of the x38 term.

2. Relevant equations
General solution is of the form:
y=a0+a1x+a2x2+a3x3+a4x4+a5x5+...

If you factor out the ao and a1 you will be left with two series that comprise the general solution

3. The attempt at a solution
I found the recurrence relation to be an+2=$$\frac{an(n+1)}{(n+2)(n+3)}$$

This makes getting the 6 terms a "plug and chug" exercise so I'm not going to type it all out.

But how do I come up with the generalization for the ao and a1 series? It would take forever to compute the x coefficient on the 38th power without a generalization so I'm totally stuck. It must have to do with the recurrence relation.

Please be as detailed as possible. I can't find this in my text or on Youtube and I have a final in 3 days and the adjunct teaching the class doesn't do office hours. :(

2. May 15, 2010

### vela

Staff Emeritus
I don't think you solved it correctly. You should only end up with one series because it's a first-order equation and should therefore only have one solution.

3. May 15, 2010

### filter54321

Ah. Typo.

y''-xy'=0

4. May 15, 2010

### LCKurtz

Which, of course, you don't need series to solve. Let y' - u...

5. May 15, 2010

### vela

Staff Emeritus
You need to recheck your recurrence relation. It's close, but not quite right.

To find a38, you will probably find it helpful to write out what a2, a4, and a6 equal explicitly. You should recognize a pattern that will let you write down what a38 is.