Final answer:
Equilibrium solutions for differential equations can be stable, unstable, or semistable. Stable equilibria return the system to equilibrium after a disturbance, while unstable equilibria cause the system to move away from equilibrium. Semistable equilibria exhibit both stable and unstable behaviors.
Step-by-step explanation:
Equilibrium Solutions and Stability
Equilibrium solutions for a differential equation are values of the independent variable at which the derivative of the dependent variable is zero. These solutions can be classified as stable, unstable, or semistable.
Stable Equilibrium:
A stable equilibrium solution occurs when the system returns to the equilibrium after being slightly disturbed. The graph of the solution approaches the equilibrium point as time goes to infinity. An example of a stable equilibrium is a ball resting in a bowl.
Unstable Equilibrium:
An unstable equilibrium solution occurs when the system moves away from the equilibrium point after being slightly disturbed. The graph of the solution moves away from the equilibrium as time goes to infinity. An example of an unstable equilibrium is a ball balanced on top of a hill.
Semistable Equilibrium:
A semistable equilibrium solution is a combination of both stable and unstable behaviors. The graph of the solution approaches the equilibrium point from one side and moves away from it on the other side. An example of a semistable equilibrium is a ball in a dip on a hill.
To find and classify equilibrium solutions, we can use a phase line, which summarizes the behaviors of the solutions.
Summary:
Equilibrium solutions of differential equations can be classified as stable, unstable, or semistable. A stable equilibrium returns the system to its original state after a disturbance, while an unstable equilibrium causes the system to move away from the equilibrium point. A semistable equilibrium exhibits both stable and unstable behaviors.