Parabolic Cylinder Differential Equation

Author

Eric W. Weisstein
March 22, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DifferentialEquations/ParabolicCylinderDifferentialEquation.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/ParabolicCylinderDifferentialEquation.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Regular

DSolve[y''[z]+(ν+(1/2)-(1/4)(z^2))y[z]==0,y[z],z]

FullSimplify[%]//TraditionalForm

Generalized

(a (z^2)+b z+c) y(z)+y''(z)==0

DSolve[Derivative[2][y][z]+(a*z^2+b*z+c)*y[z]==0,y[z],z]

FullSimplify[((-4 (a^(3/2))-I (b^2)+4 I a c)/(8 (a^(3/2))))]

FullSimplify[Expand[#]]&//@%

TraditionalForm[%]