Chapter 12.22, Problem 5P

By power series, solve the Hermite differential equation $y^{\prime \prime }-2x{y}^{\prime }+2py=0$

You should find an $a_0$ series and an $a_1$ series as for the Legendre equation in Section 2 Show that the $a_0$ series terminates when $p$ is an even integer, and the $a_1$ series terminates when $p$ is an odd integer. Thus for each integer $n$, the differential equation (22.14) has one polynomial solution of degree n. These polynomials with $a_0$ or $a_1$ chosen so that the highest order term is ${(2x)}^n$ are the Hermite polynomials. Find $H_0(x),H_1(x),$ and $H_2(x).$ Observe that you have solved an eigenvalue problem (see end of Section 2), namely to find values of p for which the given differential equation has polynomial solutions, and then to find the corresponding solutions (eigenfunctions).