Finding Equilibrium Points and Analyzing Stability

Consider the differential equation: $\frac{dx}{dt} = x(1 - x)$. Identify the equilibrium points and determine their stability.

When dealing with differential equations, particularly autonomous first-order equations like the one presented in this problem, identifying equilibrium points is a critical step. Equilibrium points, or steady states, occur where the rate of change of the system is zero. In this context, that means finding values of x for which the derivative of x with respect to t is zero. This often requires setting the right-hand side of the differential equation to zero and solving for x, which gives insight into the constant solutions of the system.

Once equilibrium points are identified, the next important step is to determine their stability. This involves analyzing how small perturbations to these equilibrium points evolve over time. By examining the sign of the derivative of the function on the right-hand side of the differential equation near these points, we can understand if these points are stable, unstable, or semi-stable. A stable equilibrium point will see perturbed values decay back to the equilibrium, while unstable points will see perturbations grow over time, moving the system away from the equilibrium.

In essence, the focus of this problem is on the stability analysis of equilibrium points of a simple nonlinear differential equation. Understanding these concepts is fundamental in mathematical modeling of physical, biological, and economic systems where understanding long-term behavior of solutions provides much-needed insight into system dynamics.