8.3.3 (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-α (j + 1)(j+2) (j even), 2(-a)x2 22(-a)(2-a)x4 Yeven ao 1+ + 2! 4! For s=1, aj+2=2aj j+1-α (j+2)(j+3) (j even), Yodd a1 + 3! 2(1a)x3 22(1 - a) (3-α)x5 (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). + 5! (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.)