reference request - Differential equation for a ratio of consecutive Bessel functions

I do not know how helpful this would be to you however it was very helpful to understand the physics and numerics of Bessel.
if you are studying elastic wave propagation. The solution of the differential equations of potential wave is the cylindrical Bessel:

$ <math>r^2 frac{d^2 R}{dr^2} + r frac{dR}{dr} + (r^2 - alpha^2)R = f(r)</math>$

for an arbitrary real integer number α (the '''order''' of the Bessel function). In solving problems in cylindrical coordinate systems, Bessel functions are of integer order (α = ''n''). Since this is a second-order differential equation, there must be two [[linearly independent]] solutions. My solutions use Bessel J(n,.) and Hankel H(n,.) (as previously mentioned)

The potential is assumed for each media to be:

$<math>phi=left(a_{1}J_{n}(K*r)+a_{2}*H_{n}(K*r)right)*e^{intheta}
,</math>$

$<math>psi(t) = left(a_{3}J_{n}(k*r)+a_{4}*H_{n}(k*r)right)*e^{intheta} ,</math> $

However for numerical stability:
They (ref. 1) normalize the potential for each layer and for each nth iteration The potential will have Hankel function equal to 1 at the inner radius, while Bessel J will be multiplied by Hankel at outer radius.

$<math>phi=left(a_{1}J_{n}(K*r)*H_{n}(K*r_{out})+a_{2}*frac{H_{n}(K*r)}{H_{n}(K*r_{in})}right)*e^{intheta},</math>$

$<math>psi(t)=left(a_{3}J_{n}(k*r)*H_{n}(k*r_{out})+a_{4}*frac{H_{n}(k*r)}{H_{n}(K*r_{in})}right)*e^{intheta},</math>$

For more details:
Bessel functions in wave propagation and scattering
Reference:
David C. Ricks and Henrik Schmidt, "A numerically stable global matrix method for cylindrically layered shells excited by ring forces" 1994