[Differential Equations] 01. Some Basic Mathematical Mode

 

1. Overview of the Chapter

• It introduces differential equations by illustrating fundamental ideas through two example problems.
• It also provides a classification of differential equations and an overview of their historical development.
• Differential equations have been a key area of study for over three centuries and continue to be an active field of research with many open questions.

2. Importance of Differential Equations

 Many natural phenomena are governed by mathematical relations involving rates of change.

 Equations that include derivatives are called differential equations.

 Understanding differential equations is crucial in fields such as Fluid motion, Electrical circuits, Heat dissipation, Seismic wave propagation, Population dynamics.

 A differential equation describing a physical process is called a mathematical model of that process.

 

3. Example 1: A Falling Object

1) Formulating the Differential Equation

Consider an object falling near sea level.

Define variables

 

• $t$: time (independent variable)
• $v$: velocity (dependent variable)
• $m$: mass of the object
• $g$: acceleration due to gravity (9.8 m/s²)
• $γ$: drag coefficient (resistance due to air)

Newton's Second Law states that force equals mass times acceleration.

$$F=ma$$

Acceleration is the time derivative of velocity

$$ a= \frac{dv}{dt}$$

The net force acting on the object is the sum of gravity and air resistance

$$ F=mg- γv$$

Final differential equation for velocity

$$m\frac{dv}{dt}=mg- γv$$

$\frac{dv}{dt}=9.8- \frac{v}{5} $(assuming $m$ = 10$kg$, $γ$=2$kg/s$)

 

2) Equilibrium Solution & Terminal Velocity

The equilibrium solution occurs when $ \frac{dv}{dt}=0$

 

 

 

he constant function $v(t)=49m/s$ is an equilibrium solution, representing terminal velocity.

If $v<49$, velocity increases. If $v>49$, velocity decreases.

Over time, all solutions converge to the equilibrium solution.

 

4. Direction Fields

A direction field is visual representation of solutions to a differential equation.

For an equation of the form

$$\frac{dy}{dt}=f(t,y)$$

 

• At each point (t,y), compute f(t,y).
• Draw a small line segment with slope f(t,y).
• These segments indicate the local direction of solution curves.

Key properties of direction fields

 

• They allow visualization of solution behavior without solving the equation.
• They help identify equilibrium solutions and stability.
• Computers are commonly used to generate direction fields.

 

5. Example 2: Population Growth(Field Mice & Owls)

1) Basic Growth Model (No Predators)

Assume a population of mice grows at a rate proportional to its current population

$$\frac{dp}{dt}=rp$$

p(t): population

r: growth rate constant(e.g., 0.5 per month)

 

2) Adding a Predator(Owls)

 Suppose owls hunt 15 mice per day -> 450 mice per month.

 Modify the equation to include predation

 $$\frac{dp}{dt}=0.5p-450$$

The equilibrium solution occurs when $\frac{dp}{dt}=0$

 

 

 

 

Solution behavior

 

• If p>900, population increases.
• If p<900, population decreases.

3) Comparing Examples 1&2

Falling Object (Example 1)	Population Growth (Example 2)
Solutions converge to the equilibrium velocity (terminal velocity).	Solutions either increase or decrease, moving away from equilibrium.
Gravity and drag force balance at equilibrium.	Growth and predation balance at equilibrium.

 

6. Constructing Mathematical Models

Steps for building a differential equation model

1) Identify variables (independent & dependent).

2) Choose measurement units (e.g., seconds, months).

3) Define governing principles (e.g., Newton’s laws, biological growth assumptions).

4) Translate principles into equations using physical constants or parameters.

5) Check unit consistency to ensure correctness.

6)Analyze the equation—solve it if possible or use direction fields.

 

 

Limitations of models

- The falling object model assumes no obstacles.

- The population model predicts unrealistic results (negative or excessively large numbers).

 

7. Key Takeaways

✔️ Differential equations describe rates of change in various physical and biological processes.
✔️ Mathematical models help analyze real-world systems, from falling objects to population growth.
✔️ Direction fields provide a powerful visual method to understand solutions.
✔️ Equilibrium solutions help determine long-term behavior.
✔️ Building a differential equation model requires careful consideration of variables, governing principles, and units.