Review: First Order Differential Equations

The simplest type of differential equation is a first order differential equation. These are equations that involve an unknown function and its first derivative. We will explore some simple examples of first order differential equations.

The Most Important Differential Equation: Exponential Growth

The most important differential equation in the world is the equation for exponential growth. This has already been covered in depth previously, so we will only briefly cover it here.

The equation for exponential growth is:

where is the function we are trying to solve for, and is a constant. By solving this differential equation, we can find the function that describes exponential growth:

The exponential function has a few interesting properties:

1. .
2. is always positive and never zero.
3. .
4. If , then grows exponentially; and .
5. If , then decays exponentially; and .
6. If , grows much faster than any polynomial function; and .

Separable Differential Equations

A separable differential equation is a first order differential equation that can be written in the form:

This equation can be solved by separating the variables and integrating both sides.

Example Problem: Constructing and Solving a DE

A population of cows grows at a rate proportional () to the number of cows present. Additionally, a farmer harvests a fixed number of cows () each year.

1. Model and write a differential equation for the population of cows.
2. Given , find the doubling time of the population of cows; i.e., the time it takes for the population to double.
3. Solve the differential equation.
4. There is a population where the growth rate is exactluy equal to the harvest rate, thus the population remains constant (at equilibrium). Find the equilibrium population.
5. Interpret all the results. What do they mean in the context of the problem?

Step 1: Creating a Model and Differential Equation

We can introduce some variables to model the population of cows:

1. : the population of cows at time , in some unit .
2. : the growth rate of the cows, in .
3. : the number of cows harvested each year, in .
4. : the initial population of cows, in .

Let's first introduce the growth term:

is the rate of change of the population of cows. Since we count per year, the unit of is cows per year. Next, we introduce the harvesting term:

Notice that this is constant unlike the term. This means that the harvest amount is constant and does not depend on the population of cows.

Step 2: Finding the Doubling Time

To find the doubling time, we set . Then, the differential equation becomes:

The doubling time is a macroscopic property that cannot be found just through the infinitesimal information provided by the differential equation. Thus we need to solve the differential equation to find the doubling time. Using separation of variables, we can solve this differential equation:

To find the doubling time, we need to find the time it takes for the population to double:

Substitute and solve:

Note that the doubling time only depends on the growth rate . Additionally, the doubling time is only used only when , that is, when the population is growing.

If the population is shrinking, the doubling time is negative and does not carry any physical meaning. However, the concept of a halving time can be defined in the same way. In this case, we can use the same equation and find that , a positive number, is the halving time.

Step 3: Solving the Differential Equation

First, we need to construct the initial condition.

If ,

Then, at , , where is the initial population.

In the case where , we can perform separation of variables to solve this differential equation:

Next, we perform a change of variables. Let , and :

Giving us the solution to the differential equation. We can find the value of by using the initial condition :

Substitute this back into the solution:

Step 4: Finding the Equilibrium Population

In the constant/equlibrium case, the population remains constant. That is to say,

Substitute this into the differential equation:

If , then the only situation where the population remains constant is when , i.e.

If , we can use two methods to obtain the equilibrium population:

1. Just using the infinitesimal information from the differential equation, we can rearrange the equation to find the equilibrium population:

2. Using the solution. The derivative of the solution is . Setting that to be :

   Then, plugging this back into the solution:

   This is the same result as the first method, but looks at it from a "macroscopic" perspective.

We can also make intuitive sense of this result. First, ensure that units match up:

Next, we can rearrange the result to get:

This should make sense as well. Our population increases at cows per year, and we harvest cows per year. Therefore, for the population to remain constant, the growth rate must be equal to the harvest rate. In other words, there should be as many cows being born as there are being harvested.

Step 5: Interpretation

First, let's consider some conditions for the model to be valid:

1. Both and must be positive.
2. The initial population must be positive and less than the equilibrium population .

However, these are just condiitons for this specific model to be valid. Hence it is still important to consider what happens when these conditions are not met.

For :

1. For populations above , the growth will outpace the harvest, and the population will grow exponentially to infinity.
2. For populations below , the growth will be outpaced by the harvest, and the population will decay exponentially to zero.

This means that when pertubed slightly, the population will deviate from the constant solution rather than return to it. A constant solution like this is known as an unstable equilibrium.

Next, consider the case where . The population will decay to zero, in which case the population will be extinct. Let be the time it takes for the population to reach zero.

Then, the time it takes for the population to reach zero is:

A plot of the population over time is shown below: