📚 Topic Summary
Equilibrium points, also known as critical points or stationary points, are solutions to a differential equation where the rate of change is zero. In other words, if your system starts at an equilibrium point, it will theoretically stay there forever. To find these points, you set the derivative (the function representing the rate of change) equal to zero and solve for the variable(s). Understanding their stability (whether solutions nearby converge towards or diverge away from them) is crucial for predicting the long-term behavior of the system. These points are critical in analyzing the behavior of dynamical systems.
🧠 Part A: Vocabulary
Match each term with its correct definition:
- Term: Equilibrium Point
- Term: Stability
- Term: Differential Equation
- Term: Asymptotically Stable
- Term: Unstable
- Definition: A solution to a differential equation where the rate of change is zero.
- Definition: The tendency of solutions near an equilibrium point to either converge towards or diverge away from it.
- Definition: An equation that relates a function with its derivatives.
- Definition: An equilibrium point where nearby solutions converge towards it as time goes to infinity.
- Definition: An equilibrium point where nearby solutions diverge away from it.
(Match the numbers to the numbers. Example: 1-A, 2-B, etc.)
✍️ Part B: Fill in the Blanks
Complete the following paragraph with the correct terms:
An ___________ point is a solution to a differential equation where the derivative equals __________. To determine the __________ of an equilibrium point, we analyze the behavior of solutions __________ to it. An equilibrium point is considered ___________ if solutions move away from it.
(Word Bank: zero, stability, equilibrium, unstable, close)
🤔 Part C: Critical Thinking
Consider the differential equation $\frac{dy}{dt} = y(y-2)(y+3)$.
- 🌱 Find all the equilibrium points.
- 📈 Determine the stability of each equilibrium point (stable or unstable). Explain your reasoning using a phase line diagram or other method.