This paper addresses the Cauchy problem for the Modified Biharmonic Equation (MBE). We start by presenting its derivation from the Stokes equations under specific assumptions and boundary conditions. A factorization technique is introduced to manage this fourth-order Cauchy problem, leading to the development of an iterative alternating method for solving the resulting second-order Cauchy problems. The finite element method is used for numerical discretization. The main findings are summarized and analyzed in the context of numerical experiments, with a discussion on their significance. Finally, we outline potential extensions and the limitations of this study.