  This work deals with the obtaining of solutions of first and second order
Stieltjes differential equations. We define the notions of Stieltjes derivative
on the whole domain of the functions involved, provide a notion of n-times
continuously Stieltjes-differentiable functions and prove existence and
uniqueness results of Stieltjes differential equations in those spaces. We also
present the Green's functions associated to the different problems and an
application to the Stieltjes harmonic oscillator.
