Bessel type differential equation

[adpc]

Homework Statement

Hello I am trying to solve the following Differential Equation:

$$r^2\frac{d^2R}{dr^2}+r\frac{dR}{dr}-\left[A^2r^4-B^2r^2-C^2]R=0$$

where A,B and C are constants-

Homework Equations

I have read this equation is calle "Bessel wave eq" but I can't find the reference which is Moon and Spencer "Handbook"

The Attempt at a Solution

So with the change of variable z=r^2 and I get the following

$$4z^2\frac{d^2R}{dz^2}+4z\frac{dR}{dz}-\left[A^2z^2-B^2z-C^2]R=0$$

which still doesn't have the form of a Bessel equation, because of the B^2z term!
How I get rid of this term? What change of variable can I make to get a Bessel equation?
Is there another way to solve this ODE?

 

[JJacquelin]

What change of variable can I make to get a Bessel equation?

I should be surprised if this equation could be transformed into a Bessel equation.
Rather it could be changed into a Kummer equation (confluent hypergeometric)

 

[JJacquelin]

Similarly : equation (5) in
http://mathworld.wolfram.com/WhittakerDifferentialEquation.html