Covering the method of solving this is probably well beyond what could be presented in a small step-by-step solutions window.

You mean to derive both of these are independant solutions:

-Sqrt[a] b Sqrt[E^(-(x/b))] BesselI[1, 2 Sqrt[a] b Sqrt[E^(-(x/b))]]

Sqrt[a] b Sqrt[E^(-(x/b))]BesselK[1, 2 Sqrt[a] b Sqrt[E^(-(x/b))]]

Are you familiar with BesselI and BesselK? Understanding the differential equations behind these special functions would be where I start.