Parabolic Cylinder Differential Equation

The parabolic cylinder differential equation is the second-order ordinary differential equation

(1)

whose solution is given by

(2)

where  is a parabolic cylinder function.

The generalized parabolic cylinder differential equation is the differential equation of the form

(3)

(Abramowitz and Stegun 1972, p. 686; Zwillinger 1995, p. 414; Zwillinger 1997, p. 126) whose solution can be expressed in terms of parabolic cylinder functions as

(4)

where

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). "Parabolic Cylinder Function." Ch. 19 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 685-700, 1972.

Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 414, 1995.

Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 126, 1997.