Self-assessment: Can you construct first-order ODEs from given physical conditions?

📚 What is a First-Order ODE?

A first-order ordinary differential equation (ODE) is an equation that involves a function and its first derivative. It describes the relationship between a quantity and its rate of change. These equations are fundamental in modeling various phenomena in physics, engineering, biology, and economics.

📜 Historical Context

The development of ODEs is closely tied to the birth of calculus in the 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Newton used differential equations to describe motion and gravity, laying the foundation for classical mechanics. Since then, ODEs have become indispensable tools for scientists and engineers.

🔑 Key Principles for Constructing First-Order ODEs

• 🔍 Identify Variables: Determine the independent and dependent variables. For example, time ($t$) is often the independent variable, and a quantity like population ($P$) or temperature ($T$) might be the dependent variable.
• 🍎 Define Relationships: Understand the physical laws or principles governing the situation. For instance, Newton's Law of Cooling states that the rate of change of an object's temperature is proportional to the difference between its own temperature and the ambient temperature.
• 📝 Translate into Math: Express the relationships mathematically using derivatives. If $P$ represents population, then $\frac{dP}{dt}$ represents the rate of change of the population with respect to time.
• 🧮 Formulate the Equation: Combine the terms to create the first-order ODE. For example, Newton's Law of Cooling can be written as $\frac{dT}{dt} = k(T - T_a)$, where $T$ is the object's temperature, $T_a$ is the ambient temperature, and $k$ is a constant of proportionality.
• 💡 Define Initial Conditions: To find a unique solution to the ODE, you need an initial condition. This is a value of the dependent variable at a specific value of the independent variable (e.g., $T(0) = T_0$, where $T_0$ is the initial temperature).

🌍 Real-World Examples

Radioactive Decay

The decay of a radioactive substance is proportional to the amount of the substance present.

• ☢️ Variables: Let $N(t)$ be the amount of radioactive substance at time $t$.
• ⚛️ Relationship: The rate of decay is proportional to $N(t)$, so $\frac{dN}{dt} = -λN(t)$, where $λ$ is the decay constant.
• 🧪 Equation: $\frac{dN}{dt} = -λN$.

Population Growth

The growth of a population is often modeled as being proportional to the current population size.

• 👨‍👩‍👧‍👦 Variables: Let $P(t)$ be the population size at time $t$.
• 🌱 Relationship: The rate of growth is proportional to $P(t)$, so $\frac{dP}{dt} = rP(t)$, where $r$ is the growth rate.
• 📈 Equation: $\frac{dP}{dt} = rP$.

Mixing Problems

Consider a tank containing a solution with a certain amount of solute. A solution with a different concentration enters the tank, and the mixture is drained.

• 💧 Variables: Let $A(t)$ be the amount of solute in the tank at time $t$. Let $V(t)$ be the volume of liquid in the tank at time $t$.
• 🌊 Relationship: $\frac{dA}{dt} = (rate\, in) - (rate\, out)$. If the concentration of solute entering is $C_{in}$ and the flow rate is $Q_{in}$, then the rate in is $C_{in}Q_{in}$. If the flow rate out is $Q_{out}$, and we assume perfect mixing, the concentration leaving is $\frac{A(t)}{V(t)}$. So, the rate out is $\frac{A(t)}{V(t)}Q_{out}$.
• ⚗️ Equation: $\frac{dA}{dt} = C_{in}Q_{in} - \frac{A(t)}{V(t)}Q_{out}$. If $V(t)$ is constant, this simplifies the equation.

📝 Practice Quiz

1. 🌡️ A cup of coffee initially at 90°C is placed in a room at 20°C. The temperature of the coffee cools at a rate proportional to the difference between its temperature and the room temperature. Write the differential equation for the temperature $T(t)$ of the coffee at time $t$.
2. 🏦 A bank account earns interest continuously at a rate of 5% per year. If an initial deposit of $1000 is made, and no further deposits or withdrawals are made, write the differential equation for the amount of money $A(t)$ in the account at time $t$.
3. 🐟 A population of fish in a lake grows at a rate proportional to the number of fish present. However, fish are also harvested at a constant rate $H$. Write the differential equation for the population $P(t)$ of fish at time $t$.
4. 🦠 A certain population of bacteria is known to grow at a rate proportional to the amount present. After an hour, the population has doubled. After two hours, the population has tripled. What was the initial population? (Hint: set up and solve the IVP).
5. 🧪 A tank contains 1000 liters of water with 10 kg of salt dissolved in it. Water containing 0.01 kg of salt per liter is pumped into the tank at a rate of 5 liters per minute, and the mixture is pumped out at the same rate. Write the differential equation for the amount of salt $S(t)$ in the tank at time $t$.

🏁 Conclusion

Formulating first-order ODEs from physical conditions requires a solid understanding of the underlying principles and careful translation of those principles into mathematical language. With practice, you'll become adept at constructing these equations and using them to model the world around you.