Find the power series in x for the general solution of (1+2x^2)y"+7xy'+2y=0

[Math10]

Homework Statement

Find the power series in x for the general solution of (1+2x^2)y"+7xy'+2y=0.

Homework Equations

None.

The Attempt at a Solution

I'll post my whole work.

 

[Math10]

This is my work, I have more to post.

 

[Math10]

This work comes first, the above one comes after this one.

 

[Math10]

But I don't know how to get to the book's answer.

 

[pasmith]

You have the correct recurrence relation $$a_{2n} = - \frac{2n+1}{n+1}a_n.$$ You just haven't tried to solve it.

Recurrences of the form $$a_{n+1} = f(n)a_n$$ have the solution $$a_n = a_0\prod_{k=0}^{n-1} f(k)$$ where by convention $$\prod_{k=0}^{-1} f(k) = 1.$$
Recurrences of the form $$a_{n+2} = f(n)a_n$$ can be turned into the above form by treating even and odd terms separately: First set $n = 2m$ and $b_m = a_{2m}$ to obtain $$b_{m+1} = f(2m)b_m$$ and then set $n = 2m+1$ and $c_m = a_{2m+1}$ to obtain $$c_{m+1} = f(2m+1)c_m.$$

 

[Math10]

But that's not the answer in the book. How do I get the answer in the book?

 

[pasmith]

Are you saying that $$y(x) = \sum_{n=0}^\infty a_nx^n = \sum_{m=0}^\infty b_mx^{2m} + \sum_{m=0}^\infty c_mx^{2m+1}$$ with $b_m$ and $c_m$ obtained as I have suggested is not the answer in the book?

What do you get for $b_m$ and $c_m$?