The Solution of the Eigenvalue Problem $\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + \lambda y = 0$

Multiple Choice

The solution of the eigenvalue problem $\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + \lambda y = 0$ where $y$ is bounded on $[ - 1,1 ]$ , is

A) $\lambda = n , y = P _ { n } ( x ) , n = 1,2,3 ,$
B) $\lambda = n - 1 , y = P _ { n } ( x ) , n = 1,2,3 , \ldots$
C) $\lambda = n + 1 , y = P _ { n } ( x ) , n = 1,2,3 , \ldots$
D) $\lambda = n ^ { 2 } , y = P _ { n } ( x ) , n = 1,2,3 , .$
E) $\lambda = n ( n + 1 ) , y = P _ { n } ( x ) , n = 1,2,3 , \ldots$