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Topic: Extension to parabolic cylinder functions?
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Leslaw Bieniasz

Posts: 173
Registered: 12/13/04
Extension to parabolic cylinder functions?
Posted: Feb 9, 2012 6:15 AM
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The so-called parabolic cylinder functions are known to represent
the solutions of the ordinary differential equation

d^2 u(x)/ dx^2 -(a*x^2 +b*x + c) * u(x) = 0

When a = 0 the equation reduces to the Airy equation, with
Airy functions as solutions.

I am trying to solve and equation with a fourth order polynomial
in the place of the second order one:

d^2 u(x)/ dx^2 -(a*x^4 + b*x^3 + c*x^2 + d*x + e) * u(x) = 0

My question is: are there any extensions of the parabolic
cylinder functions to the above equation? Do there exist
any special functions that represent solutions of the equation?

I would appreciate any pointers.


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