Hi, I am trying to solve the modified Bessel ODE of real variables:
DSolve[u''[r] + u'[r] / r - a * u[r] ⩵ 0, u[r], r, Assumptions → {a ≥ 0}]
but I obtain the result in terms of the BesselJ and BesselY functions of imaginary arguments:
u[r] → BesselJ[0, ⅈ a^1/2 r] C[1] + BesselY[0, -ⅈ a^1/2 r] C[2].
However, I would like to verify that the solution is, in fact, a linear combination of the BesselI and BesselK functions of real arguments. Calling FunctionExpand[BesselJ[0, ⅈ a^(1/2) r] gives indeed BesselI[0, a^(1/2) r], but I cannot find a way to prove that the second component, BesselY[0, -ⅈ a^1/2 r] , would be equivalent to BesselK[0, a^1/2 r], or would contribute such a term. FunctionExpand does not give such a result. Is there any working method to obtain the desired result? Lesław.