Stability Analysis of Equilibrium Solutions in Differential Equations

For a differential equation $\frac{dy}{dt} = f(y)$, determine the stability of the equilibrium solutions using the phase line and the graph of $f(y)$. Define if each equilibrium solution is asymptotically stable, unstable, or semi-stable.

In this problem, we delve into the stability analysis of equilibrium solutions for the differential equation dy/dt = f(y). At the core of this analysis is the concept of equilibrium solutions, which are the points where dy/dt equals zero. Understanding these equilibrium points is crucial because they signify the states where the system remains unchanged over time unless perturbed. The stability of these points informs us about the behavior of the system in the vicinity of the equilibrium. That is, whether solutions starting near an equilibrium point remain close to, or diverge from, the equilibrium as time progresses is determined by their stability property. To assess the stability of equilibrium points, we utilize the phase line and the graphical representation of f(y). The phase line is a powerful analytical tool that helps visualize the direction of change (increase or decrease) in y as time progresses. On this line, equilibrium points are marked, and the behavior around these points is indicated by the direction arrows. If solutions tend to move away from an equilibrium point, it is labeled as unstable; if they move towards it, it is asymptotically stable; and if solutions that are above it move downwards and solutions below move upwards, it is termed semi-stable. The graph of f(y) offers a visual verification. The slope at the equilibrium points, the graphical behavior of f(y) near these points, also plays a critical role in determining stability. As you explore these concepts, consider the implications they have for real-world systems modeled by such differential equations. Stability analysis isn't just a theoretical exercise but a practical method for predicting the behavior of physical, biological, or economic systems, and thus, a fundamental skill in applied mathematics.