(a) When a = n is a nonnegative integer Hermite's differential equation always possesses
Chapter 5, Problem 28(choose chapter or problem)
(a) When a = n is a nonnegative integer Hermite's differential equation always possesses a polynomial solution of degree n. Use y1(x) given in to find polynomial solutions for n = 0, n = 2, and n = 4. Then use Y2(x) in to find polynomial solutions for n = 1,n = 3, andn = 5. (b) A Hermite polynomial Hn(x) is defined to be an nth degree polynomial solution ofHennite' s differential equation multiplied by an appropriate constant so that the coefficient of x n in Hn(x) is 2n. Use the polynomial solutions found in part (a) to show that the first six Hermite polynomials are H0(x) = 1 Hi(x) = 2x H2(x) = 4x2 - 2 H3(x) = 8x3 - 12x Hix) = 16x4 - 48x2 + 12 H5(x) = 32x5 - 160x3 + 120x.
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