Several models of physical systems come in the form of differential equations. The main advantage of these models lies in their flexibility through the specifications of initial and/or boundary conditions or forcing functions. Although several physical models result in partial differential equations, there are also several important cases in which the models can be reduced to ordinary differential equations. One major class involves dynamic models (i.e., time-varying systems) in which the only independent variable is the time variable, known as initial value problems. Another case is when only one of the spatial dimensions is the only independent variable. For this case, it is possible that boundary conditions are specified at different points, resulting in multiple-point boundary value problems.
There are four chapters included in this part of the book to handle the analytical solutions, numerical solutions, qualitative analysis, and series solutions of ordinary differential equations. Chapter 6 discusses the analytic approaches to solving first- and second-order differential equations, including similarity transformation methods. For higher order linear differential equations, we apply matrix methods to obtain the solutions in terms of matrix exponentials and matrizants. The chapter also includes the use of Laplace transforms for solving the high-order linear ordinary differential equations.