Many real-world problems involve several quantities changing together. This section introduces systems of differential equations, starting with linear systems and progressing to nonlinear ones, providing tools to understand and analyze how multiple variables interact dynamically.
Up to now, we’ve mostly dealt with differential equations one at a time—one equation, one unknown function. But many real-world situations don’t work that way. Populations interact, chemicals react, and mechanical parts move together. To describe these systems, we need more than a single equation—we need a system of differential equations.
A system is simply a collection of differential equations that must be solved together because their unknowns are linked. Some systems are “uncoupled,” meaning that each equation can be solved independently. Others are “coupled,” where the variables feed into each other’s equations and evolve together.
In this chapter, we’ll build the foundations for working with systems. We’ll start by looking at simple cases, then move into coupled systems and see how tools like the phase plane help us visualize the relationships between variables. By the end, you’ll have the groundwork needed to understand and solve first-order linear systems.
