Solution to Legendre equation in trig form

[Jesssa]

hey guys,

I've been trying to solve this question,

http://img515.imageshack.us/img515/2583/asfj.jpg

so the general solution would be

y(cos(theta)) = C Pn(cos(theta)) + D Qn(cos(theta)) right?

and since n = 2 in this case

y(cos(theta)) = C P_2 (cos(theta)) + D Q_2 (cos(theta))

and 0<= theta < 2Pi

But when theta = 0, cos(theta) = 1 and Q_2 is undefined, so D = 0,

so y(cos(theta)) = C P_2 (cos(theta)) = (C/2) (cos^2(theta) - 1)

but

y(cos(0))=y(1) = C P_2(1) = C = ?

So would the 2pi periodic solution be

y(cos(theta)) = (C/2) (cos^2(theta) - 1) ?

Thanks

 

[vela]

You should have $P_2(x) = \frac{1}{2}(3x^2 - 1)$, but otherwise your work looks fine.