Smoothness of Polynomials With Respect to Boundary Conditions As Solutions of Ordinary Differential Equations

In this paper, we investigate special polynomial solutions of linear ordinary differential equations as functions of certain boundary conditions. We first look at solutions of y'' = 0 and y''' = 0, and observe the smooth dependence of the solutions on the boundary conditions by computing partial derivatives. Then, we characterize these partial derivatives according to their own respective boundary conditions. We also establish some relationships among these partial derivatives. In our generalization, we consider (n -􀀀 1)st degree polynomials as solutions of the boundary value problems of linear ordinary differential equations of nth derivative. We established these partial derivatives as (n -􀀀 1)st degree polynomials and satisfy their own respective boundary conditions. Finally, we discuss possible applications of the results of this paper through the modeling of hybrid phenomena with dynamic equations on time scales, and we mention extensions of the results to nonlinear differential equations.