We study a one–dimensional Helmholtz-type equation with a variable coefficient k(x) . The problem is formulated as a Volterra–Stieltjes integral equation, allowing the inclusion of singular measures beyond the continuous setting. An equivalence is obtained between weak formulations, integral representations, and derivative identities, providing an analogue of a result of Bird–Ngai–Teplyaev in the variable-coefficient setting. These results extend known analysis of fractal and measure-based Laplacians to variable-coefficient Helmholtz operators.