# System of differential equations - disagreement with paper

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!

• I don't think there is enough information here to understand what the discrepancy is between the solution you have and the solution you are trying to reach. – Chappers Sep 25 '15 at 23:48
• Mistakes do happen, your solution might be right. – vonbrand Sep 26 '15 at 1:11
• Can't you just write this system of differential equations as one matrix differential equation $[A',B']^T=M[A,B]^T$, where $M$ is a constant matrix, such that you can use eigenvalues and eigen vectors in order to find the general solution? – Kwin van der Veen Sep 26 '15 at 2:05
• @fibonatic: I do not think this is possible, since the prefactors in the equations are functions of $r$: ($\frac{m+1}{r}$ and $\frac{1}{r}$. – Dimitris A. Sep 26 '15 at 7:46
• @ Chappers: The problem is that I have the solution as: $A=K_m$ and $B=\frac{1}{\epsilon-1} f(A,A')$ which comes from (1). In the paper however, $A\propto (\epsilon+1)K_m$ and $B \propto K_{m+1}$. But I cannot see how one can derive these prefactors in the paper's solutions. They do not follow from the equations (1) or (2). – Dimitris A. Sep 26 '15 at 7:46