Chapter 11.3, Problem 9P

Problems

Problems 8-10 deal with Hermite’s equation:

$y^{″}-2x{y}^{\prime }+2\alpha y=0$, $-\infty <x<\infty$ $\left(\mathrm{11.3.13}\right)$

Show that if $\alpha =N$, a positive integer, the Equation $\left(\mathrm{11.3.13}\right)$ has a polynomial solution. Determine the polynomial solution. Determine the polynomial solutions when $\alpha =0,1,2\text{and}\text{3}$.