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How to obtain the right solution for modified Bessel ODE?

Posted 2 months ago
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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.

Maybe:

 FullSimplify[BesselY[0, -I Sqrt[a] r] // FunctionExpand // Re, 
  Assumptions -> {a > 0, r > 0}]

 (*-((2 BesselK[0, Sqrt[a] r])/\[Pi])*)
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