$e^{mx}$ in solving second order differential equations

In a book I am reading on differential equations, the author writes about the solution to a homogenous, linear, second order differential equation with constant coefficients. The author says something like, "Let us suppose that the solution is of the form $y=e^{mx}$ " . After this the author introduces the characteristic equation of a differential equation of the form mentioned above, and proceeds to describe how to solve it w/ undetermined coefficients, variation of parameters, etc.

How did mathematicians first come up with this "assumption" that the solution was of the form $y=e^{mx}$? And, are there any other forms of solutions for these types of equations?