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Parabolic Cylinder Differential Equation

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

y^('')+(nu+1/2-1/4z^2)y=0
(1)

whose solution is given by

y=c_1D_nu(z)+c_2D_(-nu-1)(iz),
(2)

where D_nu(z) is a parabolic cylinder function.

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

y^('')+(az^2+bz+c)y=0
(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

y=c_1D_(p_-)(1/2(i+1)(b+2az)a^(-3/4)) +c_2D_(p_+)(1/2(i-1)(b+2az)a^(-3/4)),
(4)

where

p_+/-=((+/-4ac∓b^2)i-4a^(3/2))/(8a^(3/2)).
(5)

See also

Parabolic Cylinder Function, Parabolic Cylindrical Coordinates

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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.

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Parabolic Cylinder Differential Equation

Cite this as:

Weisstein, Eric W. "Parabolic Cylinder Differential Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ParabolicCylinderDifferentialEquation.html

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