MATH 422/522 - Spring 2016 HW # 6 1. The differential equation d2y dy x 2 + (1 x) + ny = 0 dx dx is known as Laguerre's equation. a) Obtain the regular solution in the form ... (n k + 1) k k n( n 1)( n 2) y1 ( x) = B0 1 + (1) x 2 (k !) k =1 b) Show that this solution is a polynomial of degree n when n is a nonnegative integer, and verify that the choice B0 = 1 leads to the Laguerre polynomial of degree n, with the definition Ln ( x) = ( 1 Where ( ) n k n 1 2 x ( x) n n x ) 1! + ( 2 ) 2! ... + n! represents the binomial coefficient , n!/ ( n k )!k ! . d2y dy 2 x + 2ny = known as Hermite's equation. 0 is 2. The differential equation 2 dx dx Obtain the general solution in the form = ci un ( x ) + c2vn ( x) , where y ( x) n x 2 n(n 2) x 4 n(n 2)(n 4) x 6 . un ( x ) = 1 + + 1 1! 1 . 3 2! 1.3 .5 3! . . and n 1 x3 (n 1)(n 3) x5 (n 1)(n 3)(n 5) x 7 vn ( x) = x + + 3 1! 3.5 2! 3 .5 .7 3! [Hence verify that the solution or zero, whereas un ( x) is a polynomial of degree vn ( x) is a polynomial of degree H n ( x) .] . n when n is a positive even integer n when n is a positive odd integer. That multiple of the nth-degree polynomial for which the coefficient of and is often denoted by . . . xn is 2n is called the nth Hermite polynomial