How to turn BesselJ and BesselY into BesselI and BesselK?

Questions
1

Mathematica gives the solution of

(4 R^2 + Rv^2) Cs1[R, T] - 4 R (
\!\(\*SuperscriptBox[\(Cs1\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[R, T] + R 
\!\(\*SuperscriptBox[\(Cs1\), 
TagBox[
RowBox[{"(", 
RowBox[{"2", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[R, T])== 0

with

BesselJ[Rv/2, -I R] C[1] + BesselY[Rv/2, -I R] C[2]

How to turn the solution as above into BesselI and BesselK since the equation is the modified Bessel equation.

2

BesselJ[Rv/2, -I R] C[1] + BesselY[Rv/2, -I R] C[2] // FunctionExpand

(*-((2 (-I R)^(-Rv/2) R^(Rv/2) BesselK[Rv/2, R] C[
   2])/\[Pi]) + (-I R)^(-Rv/2) R^(-Rv/2)
   BesselI[Rv/2, 
   R] ((-I R)^Rv C[1] + (-I R)^Rv C[2] Cot[(\[Pi] Rv)/2] - 
    R^Rv C[2] Csc[(\[Pi] Rv)/2])*)
3

I am curious about why Mathematica give the modified Bessel equation the solution of Bessel functions rather than modified Bessel functions?

In[53]:= eq15 = 
 DSolve[y''[x] + 1/x y'[x] - (1 + v^2/x^2) y[x] == 0, y[x], x]

Out[53]= {{y[x] -> BesselJ[v, -I x] C[1] + BesselY[v, -I x] C[2]}}

 BesselI[Rv/2, R] C[1] + BesselK[Rv/2, R] C[2]

is the solution indeed of modified Bessel equation.

4

EQ = y’‘[x] + 1/x y’[x] - (1 + v^2/x^2) y[x] == 0;
SOL = DSolve[EQ, y, x]
{{y → Function[{x}, Evaluate@(SOL[[1, 1, 2, 2]] // FunctionExpand)]}}
EQ /. SOL // FullSimplify(We can check is solution is True)
(True)

Yes modified Bessel function are solution for this equation.

“Why Mathematica give the modified Bessel equation the solution of Bessel functions rather than modified Bessel functions?”, I do not know the answer to this question.