ABSTRACT
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.1 Introducing...The Non-Homogenous Cauchy Problem (Non-CP) . . . . . . . . . . . . . 129 5.2 Carefully Examining the One-Dimensional Version of (Non-CP) . . . . . . . . . . . . . 131
5.2.1 Solving (Non-CP)—Calculus Based . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.2.2 Solving (Non-CP)—Numerics Based . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.2.3 Building an Existence Theory for One-Dimensional (Non-CP) . . . . . . 134 5.2.4 Defining What is Meant By a Solution of (Non-CP) . . . . . . . . . . . . . . . . . 136
5.3 Existence Theory for General (Non-CP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.3.1 Constructing a Solution of (Non-CP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.3.2 Computing with the Variation of Parameters Formula . . . . . . . . . . . . . . 137 5.3.3 An Existence-Uniqueness Theorem for (Non-CP) . . . . . . . . . . . . . . . . . . . . 142
5.4 Dealing with a Perturbed (Non-CP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 5.5 What Happens to Solutions of (Non-CP) as Time Goes On and On and On...? 150
Many of the models introduced in Chapter 3 illustrate what happens when external forces of various kinds are introduced into the description of phenomena involving a system of linear ODEs or a higher-order linear differential equation. For example, the DEs arising in the beer blending model (3.13), the vertical spring model (3.17), the projectile motion model (3.42), and the 5-story floor displacement model (3.8.1) all involve external forces. As in Chapter 4, our present goal is to develop a single framework that subsumes these IVPs as special cases. The theory developed is a formal extension of the theory formulated in Chapter 4.
