Parabolic cylinder functions and parabolic functions

Abstract

The parabolic cylinder functions may, in general, be considered as solutions of the differential equation

$$\frac{{{d^2}y}}{{d{x^2}}} + (a{x^2} + bx + c)y = 0$$

((1))

which, by a simple change of variable, reduces to the form

$$\frac{{{d^2}y}}{{d{x^2}}} + (v + \frac{1}{2} - \frac{1}{4}{z^2}) = 0.$$

((2))

.