It is in fact just a systems of linear first-order ODEs.

$\begin{cases}-i\left[\partial_r+\dfrac{1}{r}\left(\dfrac{1}{2}-\nu\right)\right]u(r)=\pm kv(r)\\-i\left[\partial_r+\dfrac{1}{r}\left(\dfrac{1}{2}+\nu\right)\right]v(r)=\pm ku(r)\end{cases}$

$\begin{cases}\partial_ru(r)+\dfrac{1}{r}\left(\dfrac{1}{2}-\nu\right)u(r)=\pm kiv(r)\\\partial_rv(r)+\dfrac{1}{r}\left(\dfrac{1}{2}+\nu\right)v(r)=\pm kiu(r)\end{cases}$

$\therefore\partial_{rr}u(r)+\dfrac{1}{r}\left(\dfrac{1}{2}-\nu\right)\partial_ru(r)-\dfrac{1}{r^2}\left(\dfrac{1}{2}-\nu\right)u(r)=\pm ki\partial_rv(r)$

$\partial_{rr}u(r)+\dfrac{1}{r}\left(\dfrac{1}{2}-\nu\right)\partial_ru(r)-\dfrac{1}{r^2}\left(\dfrac{1}{2}-\nu\right)u(r)=\pm ki\left(-\dfrac{1}{r}\left(\dfrac{1}{2}+\nu\right)v(r)\pm kiu(r)\right)$

$\partial_{rr}u(r)+\dfrac{1}{r}\left(\dfrac{1}{2}-\nu\right)\partial_ru(r)-\dfrac{1}{r^2}\left(\dfrac{1}{2}-\nu\right)u(r)=\mp\dfrac{ki}{r}\left(\dfrac{1}{2}+\nu\right)v(r)-k^2u(r)$

$\partial_{rr}u(r)+\dfrac{1}{r}\left(\dfrac{1}{2}-\nu\right)\partial_ru(r)-\dfrac{1}{r^2}\left(\dfrac{1}{2}-\nu\right)u(r)=-\dfrac{1}{r}\left(\dfrac{1}{2}+\nu\right)\left(\partial_ru(r)+\dfrac{1}{r}\left(\dfrac{1}{2}-\nu\right)u(r)\right)-k^2u(r)$

$\partial_{rr}u(r)+\dfrac{1}{r}\left(\dfrac{1}{2}-\nu\right)\partial_ru(r)-\dfrac{1}{r^2}\left(\dfrac{1}{2}-\nu\right)u(r)=-\dfrac{1}{r}\left(\dfrac{1}{2}+\nu\right)\partial_ru(r)-\dfrac{1}{r^2}\left(\dfrac{1}{4}-\nu^2\right)u(r)-k^2u(r)$

$\partial_{rr}u(r)+\dfrac{1}{r}\partial_ru(r)+\left(k^2-\dfrac{1}{r^2}\left(\nu^2-\nu+\dfrac{1}{4}\right)\right)u(r)=0$

$r^2\partial_{rr}u(r)+r\partial_ru(r)+\biggl(k^2r^2-\left(\nu-\dfrac{1}{2}\right)^2\biggr)u(r)=0$

$u(r)=\begin{cases}C_1J_{\nu-\frac{1}{2}}(kr)+C_2Y_{\nu-\frac{1}{2}}(kr)&\text{when}~\nu-\dfrac{1}{2}\text{is an integer}\\C_1J_{\nu-\frac{1}{2}}(kr)+C_2J_{\frac{1}{2}-\nu}(kr)&\text{when}~\nu-\dfrac{1}{2}\text{is not an integer}\end{cases}$

According to http://people.math.sfu.ca/~cbm/aands/page_361.htm,

$\partial_ru(r)=\begin{cases}-C_1\left(J_{\nu+\frac{1}{2}}(kr)-\dfrac{1}{r}\left(\nu-\dfrac{1}{2}\right)J_{\nu-\frac{1}{2}}(kr)\right)-C_2\left(Y_{\nu+\frac{1}{2}}(kr)-\dfrac{1}{r}\left(\nu-\dfrac{1}{2}\right)Y_{\nu-\frac{1}{2}}(kr)\right)&\text{when}~\nu-\dfrac{1}{2}\text{is an integer}\\-C_1\left(J_{\nu+\frac{1}{2}}(kr)-\dfrac{1}{r}\left(\nu-\dfrac{1}{2}\right)J_{\nu-\frac{1}{2}}(kr)\right)+C_2\left(J_{-\nu-\frac{1}{2}}(kr)-\dfrac{1}{r}\left(\dfrac{1}{2}-\nu\right)J_{\frac{1}{2}-\nu}(kr)\right)&\text{when}~\nu-\dfrac{1}{2}\text{is not an integer}\end{cases}$

$\therefore v(r)=\begin{cases}\mp\dfrac{i}{k}\left(-C_1\left(J_{\nu+\frac{1}{2}}(kr)-\dfrac{1}{r}\left(\nu-\dfrac{1}{2}\right)J_{\nu-\frac{1}{2}}(kr)\right)-C_2\left(Y_{\nu+\frac{1}{2}}(kr)-\dfrac{1}{r}\left(\nu-\dfrac{1}{2}\right)Y_{\nu-\frac{1}{2}}(kr)\right)+\dfrac{1}{r}\left(\dfrac{1}{2}-\nu\right)(C_1J_{\nu-\frac{1}{2}}(kr)+C_2Y_{\nu-\frac{1}{2}}(kr))\right)&\text{when}~\nu-\dfrac{1}{2}\text{is an integer}\\\mp\dfrac{i}{k}\left(-C_1\left(J_{\nu+\frac{1}{2}}(kr)-\dfrac{1}{r}\left(\nu-\dfrac{1}{2}\right)J_{\nu-\frac{1}{2}}(kr)\right)+C_2\left(J_{-\nu-\frac{1}{2}}(kr)-\dfrac{1}{r}\left(\dfrac{1}{2}-\nu\right)J_{\frac{1}{2}-\nu}(kr)\right)+\dfrac{1}{r}\left(\dfrac{1}{2}-\nu\right)(C_1J_{\nu-\frac{1}{2}}(kr)+C_2J_{\frac{1}{2}-\nu}(kr))\right)&\text{when}~\nu-\dfrac{1}{2}\text{is not an integer}\end{cases}$

$v(r)=\begin{cases}\pm\dfrac{i}{k}\left(C_1J_{\nu+\frac{1}{2}}(kr)+C_2Y_{\nu+\frac{1}{2}}(kr)\right)&\text{when}~\nu-\dfrac{1}{2}\text{is an integer}\\\pm\dfrac{i}{k}\left(C_1J_{\nu+\frac{1}{2}}(kr)-C_2J_{-\nu-\frac{1}{2}}(kr)\right)&\text{when}~\nu-\dfrac{1}{2}\text{is not an integer}\end{cases}$

This post imported from StackExchange Mathematics at 2014-06-01 19:31 (UCT), posted by SE-user doraemonpaul