Solved Examples: Finding and Classifying Equilibrium Points in ODEs

📚 Quick Study Guide

• 🔍 Equilibrium Points: Equilibrium points, also known as critical points or stationary points, are the values of $y$ for which $\frac{dy}{dt} = 0$. These points represent constant solutions to the differential equation.
• 📈 Finding Equilibrium Points: To find equilibrium points, set the differential equation equal to zero and solve for $y$. For example, given $\frac{dy}{dt} = f(y)$, solve $f(y) = 0$.
• 📊 Classifying Equilibrium Points: Equilibrium points can be classified as stable, unstable, or semi-stable. This classification depends on the behavior of solutions near the equilibrium point.
• ➡️ Stability Analysis:
  • Stable: Solutions near the equilibrium point converge towards it as $t$ increases.
  • Unstable: Solutions near the equilibrium point move away from it as $t$ increases.
  • Semi-stable: Solutions approach the equilibrium point from one side but move away from it from the other side.
• 🧪 Using the Sign of $\frac{dy}{dt}$:
  • If $\frac{dy}{dt} > 0$ for $y < y_e$ and $\frac{dy}{dt} < 0$ for $y > y_e$, then $y_e$ is stable.
  • If $\frac{dy}{dt} < 0$ for $y < y_e$ and $\frac{dy}{dt} > 0$ for $y > y_e$, then $y_e$ is unstable.
  • If $\frac{dy}{dt}$ has the same sign on both sides of $y_e$, then $y_e$ is semi-stable.
• 📝 Linearization (Advanced): For more complex systems, linearization around the equilibrium point can be used to determine stability. Calculate the Jacobian matrix and analyze its eigenvalues.

Practice Quiz

1. Consider the differential equation $\frac{dy}{dt} = y(y-2)$. Which of the following are the equilibrium points?

   1. $y = 0$ only
   2. $y = 2$ only
   3. $y = 0$ and $y = 2$
   4. $y = -2$ and $y = 2$

2. For the differential equation $\frac{dy}{dt} = y - y^2$, classify the stability of the equilibrium point $y = 0$.

   1. Stable
   2. Unstable
   3. Semi-stable
   4. Cannot be determined

3. Given $\frac{dy}{dt} = (y-1)(y-3)$, classify the stability of $y=1$.

   1. Stable
   2. Unstable
   3. Semi-stable
   4. Asymptotically stable

4. Which of the following differential equations has equilibrium points at $y = -1$ and $y = 1$?

   1. $\frac{dy}{dt} = y^2 - 1$
   2. $\frac{dy}{dt} = y^2 + 1$
   3. $\frac{dy}{dt} = y - 1$
   4. $\frac{dy}{dt} = -y - 1$

5. Consider $\frac{dy}{dt} = -(y-2)^2$. Classify the equilibrium point at $y = 2$.

   1. Stable
   2. Unstable
   3. Semi-stable
   4. Asymptotically stable

6. For the equation $\frac{dy}{dt} = y^3 - 4y$, find all equilibrium points.

   1. $y = 0$ only
   2. $y = 2$ and $y = -2$ only
   3. $y = 0$, $y = 2$, and $y = -2$
   4. No equilibrium points

7. If $\frac{dy}{dt} = f(y)$ and $f'(y_e) < 0$ at an equilibrium point $y_e$, what can be said about the stability of $y_e$?

   1. $y_e$ is stable
   2. $y_e$ is unstable
   3. $y_e$ is semi-stable
   4. No conclusion can be made