 Solution

The origin is an ordinary point of the Hermite differential equation, so put $y={\sum }_{n=0}^{\infty }{a}_n{x}^n$. Substitution and matching powers givesThus the linear recurrence relation isEven and odd coefficients evolve independently from arbitrary $a_0,a_1$. The general power series solution iswith an empty product equal to one. Its first terms areBoth series converge at every finite $x$, since the ratio of successive terms in each parity subsequence tends to zero. For a nonzero polynomial of degree $N$, the highest coefficient in the differential equation requires $\lambda =2N$. Conversely, at such a value the appropriate parity series terminates, while its opposite-parity initial coefficient can be set to zero. HenceHere $H_N$ is a physicists' Hermite polynomial. Two explicit choices areDirect differentiation verifies both. These are polynomial solutions up to an arbitrary overall factor. Without a polynomial requirement or additional boundary conditions, the equation also has nonpolynomial solutions for other real $\lambda$; discreteness here comes from termination.