Power Series for ODE: Find Coefficient of x38 Term

[filter54321]

Homework Statement

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.

Homework Equations

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

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

The Attempt at a Solution

I found the recurrence relation to be a_n+2=\frac{a_n(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 a_o 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. :(

Thanks in advance.

 

[vela]

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.

 

[filter54321]

Ah. Typo.

y''-xy'=0

 

[LCKurtz]

Ah. Typo.

y''-xy'=0

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

 

[vela]

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.