Separable differential equations are those in which the dependent and independent variables can be separated on opposite sides of the equation.

Divide and Conquer

is literally true (with the mathematical definition of “divide”) in the context of differential equations. Rather than talk about math, let’s just show you what we’re getting at.

Now that we’ve seen an example, let’s discuss this in general.

**separable differential equation**is a differential equation which can be written in the form In other words, the independent variable and the function can be placed on

**separate**sides of the equals sign.

We will work with all the notation above.

The reason we care about separable differential equations is that:

- Separable differential equations help model many real-world contexts.
- Separable differential equations are solvable by humans.

The basic ideas is if then we can integrate both sides, writing

If we can symbolically compute these integrals, then we can solve for . It is now time to work some examples.

#### Proportional reasoning

In this example, we will show the power of knowing that one quantity is proportional to another.

- the radius changes at rate proportional to the snowball’s surface area, and
- the surface area is proportional to the square of the snowball’s radius.

When is the radius of the snowball inches?

We know that , so let’s put that information to use

We also know that . Again write with me, noting that

Hence To find when the radius is inches, solve and we find is hours.

- It was solved
*without*using the formula for the surface area of a sphere. For us, it was sufficient to know that the surface area of a sphere is proportional to the square of the radius. - Our solution, can be plotted in the slope field determined by :

#### Exponential and logistic growth

In the science fiction television series *Star Trek*, a **tribble** is an alien species that is
furry, spherical (radius inches), that essentially does nothing but eat and reproduce.

However we also know that . Write with me

After hours, we should have tribbles. So and we may write

Hence we can model the population growth with

Checking this with Mr. Spock’s computation, we see that

just as Mr. Spock stated.

If we graph the solution to the differential equation representing the population of tribbles that we found above

**ecological disaster**. Exponential growth can be scary.

Now we will see a model of population growth with environmental limitations.
Suppose that the birthrate of the tribbles is limited by how much food and space is
available. This gets us to the idea of *carrying capacity*.

**carrying capacity**of a biological system is the maximum population that can be sustained indefinitely with the given resources.

Let’s work an example involving this concept:

**product**and . This makes sense as when is small, the growth rate should be small; and when is small, the growth rate should also be small. Write with me Ah! This is a separable differential equation. Let’s solve it:

Use partial fraction decomposition to compute the antiderivative: However we also know that . Write with me

After hours, we should have tribbles. So and we may write

Hence we can model the population growth with

Solving for we find If we graph this solution, we see a nice “S-curve”

In our last example we used the differential equation This is called the **logistic
differential equation** where and are constants. Let’s examine a slope field for this
model with some reasonable values of and :

#### Predator-prey model

In nature there are animals that are predators and animals that are prey. In the early 20th Century, Lotka and Volterra suggested the following model to help us understand populations of predators and prey. Let

Lotka and Volterra made the following assumptions:

- If there are no prey, then the predators starve at some rate proportional to the number of predators. If there are prey, then the predator’s population grows at a rate proportional to the product of the population of the predators and the prey. In the language of calculus we write
- If there are no predators, then the prey’s population grows at some rate proportional to the number of prey. If there are predators, then the prey’s population is reduced at a rate proportional to the product of the population of the predators and the prey. In the language of calculus we write

Taking these two equations, and dividing them we find

In the previous examples, it worked out that we could solve for as a function of . As we will see with the predator-prey model, this is not always the case.

Remembering that the solution must pass through point , we write

Hence we have the following implicit equation relating and : Let’s look at this solution along with the slope field: