Not every system of differential equations needs to be solved explicitly to be understood. In this section, we focus on qualitative methods: tools that help us visualize how a system behaves, predict long-term motion, and recognize patterns like decay, spirals, and saddles without finding exact formulas. These ideas give us a big-picture view of linear systems and prepare us for deeper analysis later.
The phase plane is a two-dimensional space where we plot one variable on the horizontal axis and the other on the vertical axis. A solution to the system becomes a trajectory — a path through the plane traced as time flows.
To see the “push” of the system without fully solving it, we can draw a direction field (or slope field): at each point \((x, y)\text{,}\) we sketch a small arrow showing the vector \((dx/dt, dy/dt)\text{.}\)
In the phase plane, every trajectory moves toward the origin — trajectories are generally curved (except along the axes, which are straight-line solutions), since \(x\) and \(y\) decay independently at different rates.
For more interesting systems — like ones with partial or full coupling — the direction field can show curved paths, spirals, or saddle-shaped flows. This visual approach helps us predict system behavior even before we dive into equations.
So far, we’ve seen examples of systems in which variables evolve independently or influence one another. Now let’s step back and look at how to visualize a system as a whole.
The phase plane is a two-dimensional space where we plot one variable on the horizontal axis and the other on the vertical axis. A solution to a system like
To understand how the system behaves without solving it, we can draw a slope field or direction field: a grid of arrows showing the direction of motion at each point.
The phase portrait shows trajectories curving toward the origin (the axes themselves are straight-line trajectories), because each variable decays independently at its own rate.
This interactive slope field shows the direction of motion for the system \(x' = x + y\text{,}\)\(y' = -x + y\text{.}\) Each arrow represents the vector \((dx/dt, dy/dt)\) at that location.
Some systems push every solution toward a single point (stable equilibrium). Others send trajectories outward (unstable). Some cause spirals, as if the solution is both rotating and growing or shrinking at the same time.
Later, we’ll connect these patterns to the algebra of the system’s coefficients. But even now, these “big-picture” behaviors help us think about what the math is saying.
