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 <br /> a_{n+1} = f(n)a_n have the solution <br /> a_n = a_0\prod_{k=0}^{n-1} f(k) where by convention <br /> \prod_{k=0}^{-1} f(k) = 1.
Recurrences of the form <br /> 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 <br /> b_{m+1} = f(2m)b_m and then set n = 2m+1 and c_m = a_{2m+1} to obtain <br /> 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 <br /> 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?