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.

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])*)