Use the method of Frobenius to

[s3a]

Homework Statement

Use the method of Frobenius to find one solution near x = 0 of
x^2 y'' + (x^2 + 2x)y' - 2y = 0.

Homework Equations

Summation notation.

(The solution in my book is using the "dot dot dot" notation which I personally hate a lot since it requires writing a finite amount of infinite terms whereas the summation notation accurately represents all the infinite terms.)

The Attempt at a Solution

My attempt is attached. Sorry for the occasional bad handwriting, it's because I forget I am writing for you guys. I still think the ugly parts are legible though but I don't know if that's just me.

Can the roots of the indicial equation be different depending on the method used to solve the problem?

Assuming that they can't, the book gets roots λ_1 = 1 and λ_2 = -2 whereas I get roots r_1 = 0 and r_2 = 1.

The correct answer according to the book is:
y_1(x) = a_0 * x * Σn=0,inf (-1)^n 3! x^n /(n+3)!
y_1(x) = 3a_0/x^2 * [2 - 2x + x^2 - 2e^(-x)]

 

[kurt.physics]

My answer is:y_1(x) = a_0 * x * Σn=0,inf (-1)^n (n+2)! x^n / (2n+3)!y_1(x) = -a_0/x * [1 + x - x^2 + 2e^(-x)]Are my answers wrong or did I just get different roots? Or maybe I am confused and the book's answer is the same as mine?Thanks in advance.