This kind of equations are known to have horizontal asymptotes (as x -> infinity)
Asymptote - Wikipedia, the free encyclopedia
In this situation, your function is going to approach c/b if a > 0.
There are classes of differential equations that satisfy these kind of things and some of them are relating to heating and cooling problems in Thermodynamics and have specific DE structures.
For this problem I suggest you find the DE that satisfies this equation and then see what kind of family this DE belongs to.