Real-World Examples of First-Order Differential Equations & Their Applications

📚 Quick Study Guide

• ⏱️ First-order differential equations involve a function and its first derivative. They describe rates of change.
• 🌡️ Newton's Law of Cooling: Describes how the temperature of an object changes in relation to its environment: $dT/dt = k(T - T_s)$, where $T$ is the object's temperature, $T_s$ is the surrounding temperature, and $k$ is a constant.
• 🌱 Exponential Growth/Decay: Models population growth or radioactive decay: $dP/dt = kP$, where $P$ is the population or amount, and $k$ is a constant (positive for growth, negative for decay).
• 💧 Mixing Problems: Analyze the amount of a substance in a mixture changing over time. Example: $dA/dt = (rate\_in) - (rate\_out)$.
• सर्किट Electrical Circuits: Describes the current in a simple RL circuit: $L(dI/dt) + RI = V(t)$, where $L$ is inductance, $R$ is resistance, $I$ is current, and $V(t)$ is the voltage source.
• 💡 Integrating Factor Method: A common technique to solve linear first-order ODEs: Multiply the equation by an integrating factor $e^{\int p(t) dt}$ to make it easier to solve.

Practice Quiz

1. Which of the following real-world phenomena can be modeled by a first-order differential equation?
   1. A. The motion of a simple pendulum.
   2. B. The growth of a bacteria colony.
   3. C. The trajectory of a projectile with air resistance.
   4. D. The oscillations in an LC circuit.
2. Newton's Law of Cooling is described by the equation $dT/dt = k(T - T_s)$. What does $T_s$ represent?
   1. A. The initial temperature of the object.
   2. B. The surrounding temperature.
   3. C. The rate of cooling.
   4. D. The time elapsed.
3. The differential equation $dP/dt = kP$ models exponential growth when:
   1. A. $k < 0$
   2. B. $k = 0$
   3. C. $k > 0$
   4. D. $k = 1$
4. In a mixing problem, what does $dA/dt = (rate\_in) - (rate\_out)$ represent?
   1. A. The total amount of substance in the tank.
   2. B. The rate of change of the amount of substance in the tank.
   3. C. The volume of the tank.
   4. D. The concentration of the substance.
5. For an RL circuit, which component is represented by the term $L$ in the equation $L(dI/dt) + RI = V(t)$?
   1. A. Resistor
   2. B. Inductor
   3. C. Capacitor
   4. D. Voltage Source
6. What is the purpose of using an integrating factor when solving a linear first-order differential equation?
   1. A. To make the equation non-linear.
   2. B. To simplify the integration process.
   3. C. To find the initial conditions.
   4. D. To eliminate the derivative term.
7. A population of bacteria doubles every hour. Which differential equation best models this situation, where P(t) is the population at time t?
   1. A. $dP/dt = -P$
   2. B. $dP/dt = P^2$
   3. C. $dP/dt = P$
   4. D. $dP/dt = 2P$