On second-order linear Stieltjes differential equations with non-constant coefficients

In this work, we define the notions of Wronskian and simplified Wronskian for Stieltjes derivatives and study some of their properties in a similar manner to the context of time scales or the usual derivative. Later, we use these tools to investigate second-order linear differential equations with Stieltjes derivatives to find linearly independent solutions, as well as to derive the variation of parameters method for problems with gg-continuous coefficients. This theory is later illustrated with some examples such as the study of the one-dimensional linear Helmholtz equation with piecewise-constant coefficients.