I have tried to solve a PDE by using Laplace-Carson transformation and obtained the following solution in Laplace-Carson domain. Is it possible to put the solution in the following Modified-Bessel function?
I got this solution:
$$ y = r^β (C_1 (J_\nu(- i s r)) + C_2 (Y_\nu(- i s r)))$$
I want to reach a form like this solution.
$$ y = ((2)^{1 - β}/\Gamma\left(β\right)) (s r)^β (K_\nu( s r))$$
Knowing that:
$$Y(0) = 1$$
$$Y(Infinity) = 0$$