I have the following system of differential equations: $$A'-\frac{m}{r}A=(\epsilon-1)B \tag{1}$$ and $$B'+\frac{m+1}{r}B=-(\epsilon+1)A \tag{2}$$ where $A$ and $B$ are functions of $r$ and $A'$ and $B'$ indicate differentation with respect to $r$. $m$ and $\epsilon$ are just parameters and not functions.
I know that $\epsilon<1$. By solving (1) for $B$ and substituting in (2), I get a differential equation of second order for $A$. Its solutions are modified bessel functions of the first and second kind. Due to boundary conditions, only second kind is admissable solution. As a result, $A = K_m$. Consequently, $B$ can be found by direct substitution of the solution of $A$ in (1) and by dividing by $(\epsilon-1)$. As a result, $B=\frac{1}{\epsilon-1}...$.
Although this might seem straight-forward, in a paper, these solutions are presented as $A=(\epsilon+1)K_m$ and $B=K_{m+1}$. I have tried a lot of ways, to get to the same solutions for this system, but so far, the solutions that I have derived (which are textbook examples), cannot be transformed into those of the paper. Any helpful ideas would help me get unstuck!