(a) Develop series solutions for Hermite's differential equation y"-2xy +2ay = 0. ANS. s(S-1) = 0, indicial equation. For s = 0, aj+2=2aj j-a (j+1)(j+2) (j even), Yeven ao 1+ = [14 2(-a)x2 22(-a)(2-a)x4 + + 2! 4! For s = 1, aj+2=2aj j+1-α (j+2)(j+3) (j even), Yodd == a₁ [x + 2(1-a)x3 22(1 - a)(3-α)x5 + + 3! 5! ---]-. (b) Show that both series solutions are convergent for all x, the ratio of successive coefficients behaving, for a large index, like the corresponding ratio in the expan- sion of exp(x²). 8.4 Variation Method 395 (c) Show that by appropriate choice of a, the series solutions may be cut off and converted to finite polynomials. (These polynomials, properly normalized, become the Hermite polynomials in Section 18.1.)