Given the following equation
$$\frac{J_{n - 1} (u)}{uJ_n (u)} - \frac{K_{n-1}(w)}{wK_n(w)} = 0$$
(where $J$ is the Bessel function of the first kind, $K$ is the modified Bessel function of the second kind, $u, w \in \mathbb{R}$, $u \neq w$ in general)
it should be proved that it is equivalent to
$$\frac{J_{n - 1} (u)}{uJ_{n - 2} (u)} + \frac{K_{n-1}(w)}{wK_{n-2}(w)} = 0$$
by using some recurrence relations (as the ones presented in this document).
By comparing the two (allegedly equivalent) equations, it should be
$$\frac{J_{n - 1} (u)}{J_{n} (u)} = \frac{J_{n - 1} (u)}{J_{n - 2} (u)}$$
$$\frac{K_{n - 1} (w)}{K_{n} (w)} = - \frac{K_{n - 1} (w)}{K_{n - 2} (w)}$$
If it is right, how to proceed now?
The equality should be proved for any integer $n \geq 0$ and at least for any non-negative real values $u,v$. It is used to describe the optical fibers propagation (in documents like this, to obtain (17.104) from (17.103)).
