It is in fact just a systems of linear first-order ODEs.
{−i[∂r+1r(12−ν)]u(r)=±kv(r)−i[∂r+1r(12+ν)]v(r)=±ku(r)
{∂ru(r)+1r(12−ν)u(r)=±kiv(r)∂rv(r)+1r(12+ν)v(r)=±kiu(r)
∴∂rru(r)+1r(12−ν)∂ru(r)−1r2(12−ν)u(r)=±ki∂rv(r)
∂rru(r)+1r(12−ν)∂ru(r)−1r2(12−ν)u(r)=±ki(−1r(12+ν)v(r)±kiu(r))
∂rru(r)+1r(12−ν)∂ru(r)−1r2(12−ν)u(r)=∓kir(12+ν)v(r)−k2u(r)
∂rru(r)+1r(12−ν)∂ru(r)−1r2(12−ν)u(r)=−1r(12+ν)(∂ru(r)+1r(12−ν)u(r))−k2u(r)
∂rru(r)+1r(12−ν)∂ru(r)−1r2(12−ν)u(r)=−1r(12+ν)∂ru(r)−1r2(14−ν2)u(r)−k2u(r)
∂rru(r)+1r∂ru(r)+(k2−1r2(ν2−ν+14))u(r)=0
r2∂rru(r)+r∂ru(r)+(k2r2−(ν−12)2)u(r)=0
u(r)={C1Jν−12(kr)+C2Yν−12(kr)when ν−12is an integerC1Jν−12(kr)+C2J12−ν(kr)when ν−12is not an integer
According to http://people.math.sfu.ca/~cbm/aands/page_361.htm,
∂ru(r)={−C1(Jν+12(kr)−1r(ν−12)Jν−12(kr))−C2(Yν+12(kr)−1r(ν−12)Yν−12(kr))when ν−12is an integer−C1(Jν+12(kr)−1r(ν−12)Jν−12(kr))+C2(J−ν−12(kr)−1r(12−ν)J12−ν(kr))when ν−12is not an integer
∴v(r)={∓ik(−C1(Jν+12(kr)−1r(ν−12)Jν−12(kr))−C2(Yν+12(kr)−1r(ν−12)Yν−12(kr))+1r(12−ν)(C1Jν−12(kr)+C2Yν−12(kr)))when ν−12is an integer∓ik(−C1(Jν+12(kr)−1r(ν−12)Jν−12(kr))+C2(J−ν−12(kr)−1r(12−ν)J12−ν(kr))+1r(12−ν)(C1Jν−12(kr)+C2J12−ν(kr)))when ν−12is not an integer
v(r)={±ik(C1Jν+12(kr)+C2Yν+12(kr))when ν−12is an integer±ik(C1Jν+12(kr)−C2J−ν−12(kr))when ν−12is not an integer
This post imported from StackExchange Mathematics at 2014-06-01 19:31 (UCT), posted by SE-user doraemonpaul