The interior Dirichlet boundary value problem for the diffusion equation in non-homogeneous media is reduced to a system of Boundary-Domain Integral Equations (BDIEs) employing the parametrix obtained in \cite{carlos2} different from \cite{mikhailov1}. We further extend the results obtained in \cite{carlos2} for the mixed problem in a smooth domain with \(L^{2}(\Omega)\) right hand side to Lipschitz domains and PDE right-hand in the Sobolev space \(H^{-1}(\Omega)\), where neither the classical nor the canonical co-normal derivatives are well defined. Equivalence between the system of BDIEs and the original BVP is proved along with their solvability and solution uniqueness in appropriate Sobolev spaces.