ABSTRACT
Thus far in this text, the focus has been on individual differential equations. But there are many applications — such as the modeling of the population dynamics of multiple interacting species — that lead to systems of differential equations sharing common variables and solutions.
This chapter begins a brief, and somewhat elementary, discussion of these systems. The basic notions, terminology and notation for systems of differential equations is introduced, and it is shown both how systems naturally arise in applications, and how relatively simple first-order systems equivalent to higher-order systems and single equations can be derived. The later, it should be noted, leads to particularly illuminating tools for the analysis of single higher-order nonlinear differential equations. After this, there is a brief discussion of solving certain systems of differential equations via the Laplace transform. The chapter ends with a discourse on how the existence and uniqueness theorems for single first-order equations extend to analogous existence and uniqueness theorems for first-order systems of differential equations, and how those results, combined with the aforementioned equivalence between higher-order differential equations and first-order systems yield previously discussed existence and uniqueness theorems for single higher-order differential equations.
