2. The Hermite differential equation is H," - 2cH,' + 2nH, = 0. a) Solve this equation by series solution and show that it terminates for integral values of n . b) Use the series solution to generate the first four Hermite polynomials which are Ho(x ) = 1, H 2(x ) = 4x- 2, Hi(x ) = 2x , Ha(x ) = 8x- 12c . c) Obtain the first four Hermite polynomials using the generating function which is In g (x , t) = et'+2to = EH, (x) n ! n=0 d) Using the generating function derive the recurrence relations; Hn +1(x ) - 2cH, (x ) + 2nH, -1(x ) = 0, Hn'(x ) - 2nHn -1(2 ) = 0 . e) Using the result of part (d) verify that the H, (x ) defined by the gen- erating function obeys the Hermite differential equation