(adsbygoogle = window.adsbygoogle || []).push({}); 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 x^{38}term.

2. Relevant equations

General solution is of the form:

y=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}+a_{5}x^{5}+...

If you factor out the a_{o}and a_{1}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 a_{n+2}=[tex]\frac{a_{n}(n+1)}{(n+2)(n+3)}[/tex]

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 a_{o}and a_{1}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. :(

Thanks in advance.

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# Homework Help: Power series for ODE

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