Determining Legendre polynomials (Boas)

[benabean]

I am having trouble evaluating the Legendre Polynomials (LPs). I can do it by Rodrigues' formula but I am trying to understand how they come about.

Basically I have been reading Mary L. Boas' Mathematical Methods in the Physical Sciences, 3E. Ch.12 §2 Legendre Polynomials pg566.
In the text, she arrives at the LPs via the general solution to Legendre's differential equation and the recursion relation for evaluating coefficients. (I'd write out the formulas but apparently Latex is broken at the moment.) When I try to evaluate them by this method I cannot get the right answer;

a_0 series = series with even powers of x
a_1 series = series with odd powers of x
y = a_0y_1 + a_1y_2

For l=0, the a_1 series diverges but the a_0 series gives y=a_0
So we can write P_0(x) =1 where P is the Legendre Polynomial of the form P_l(x).

However, when I try this I get y=a_0[1] + a_1[x + O(x^3)], where O=order. ie. y=a_0 +a_1x.
Why do I have the x term? Is it because the a_1 series diverges, so I should just disregard it?

Apologies for the formatting. It is better seen if you have a copy of the book.

Any help would be appreciated
benabean.

 

[matematikawan]

Remember when solving the Legendre DE you always obtained two linearly independent solutions. One is a polynomial (finite) and the other one is an infinite series.

To me, it looks like you have solve it correctly where
y₁=1 and y₂= x + O(x^3)

Cheer!