Unstable Equilibrium Example in Differential Equations

Come up with an example of an equation that exhibits an unstable equilibrium where one arrow is going into the equilibrium and one arrow is going out.

In the realm of differential equations, the concept of equilibrium is essential for understanding the long-term behavior of systems modeled by these equations. Equilibrium points, also known as fixed points, occur where the derivative (or rate of change) equals zero. At these points, the system is in a state of balance, and no changes will occur unless disturbed. However, not all equilibria are stable. An unstable equilibrium is particularly interesting as it represents a situation where a small perturbation from the equilibrium can lead to significant changes over time.

To conceptualize an unstable equilibrium, imagine a ball resting at the top of a hill. The peak of the hill is the unstable equilibrium point. If the ball is nudged slightly, it will roll down the hill, diverging away from the equilibrium position. In mathematical terms, this is shown by having one directional arrow moving towards the equilibrium from one side and another arrow pointing away on the other, symbolizing that small perturbations are amplified.

When constructing or analyzing an equation for an unstable equilibrium, it's crucial to look at the system's stability by examining the sign and directions of the derivatives around these points. One classic example is the simple model of a pendulum, where an upright position represents an unstable equilibrium, as any slight push causes it to swing away. Such conceptual examples help in understanding complex systems in physics, biology, and economics where predicting the future state of the system is necessary for effective management and control.