How to find solutions to a Legendre equation?

[Nat3]

Homework Statement

Find the first three positive values of $\lambda$ for which the problem:

$(1-x^2)y^n-2xy'+\lambda y = 0, \ y(0)=0, \ y(x)$ & $y'(x)$ bounded on $[-1, 1]$

has nontrivial solutions.

Homework Equations

When n is even:

$y_1(x) = 1 - \frac{n(n+1)}{2!}x^2 + \frac{(n-2)(n(n+1)(n+3)}{4!} - ...$

When n is odd:

$y_2(x) = x - \frac{(n-1)(n+2)}{3!} + \frac{(n-3)(n-1)(n+2)(n+4)}{5!} - ...$

The Attempt at a Solution

I was able to do part of the problem myself. By plugging in zero into both of the above equations and then comparing to initial conditions, I was able to determine that n is odd. However, I don't know what to do next.

My textbook does not have *any* examples for this type of problem and I haven't been able to find anything I understand online. I looked the problem up on Chegg and found it here:

http://www.chegg.com/homework-help/a-first-course-in-differential-equations-with-modeling-applications-10th-edition-chapter-6.4-problem-47e-solution-9781111827052

Steps 1-2 on there are basically what I did, but then they jump to the answer in step 3 with zero explanation as to how they got there.

I don't want to just copy what they did with no understanding of how they got there, I want to actually understand it.. Can anyone explain what I need to do next to get the solutions?

Thanks :)

 

[vela]

The usual approach is to find a series solution. The requirement that y and y' remain bounded requires that $\lambda$ take on only certain values.