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Separable Differential Equation: Definition & Examples

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  • 0:04 Separable Differential…
  • 1:43 How to Solve an Equation
  • 2:24 One More Problem
  • 3:34 Lesson Summary
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Lesson Transcript
Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

A separable differential equation, the simplest type to solve, is one in which the variables can be separated. In this lesson, learn how to recognize and solve these equations.

Separable Differential Equation

Sanjay is a microbiologist, and he's trying to come up with a mathematical model to describe the population growth of a certain type of bacteria. He grows the bacteria and monitors the population over time. While doing this, he notices that the population at any given time depends on on how quickly the population is changing (the rate of change of the population). How can he represent this phenomenon mathematically and use his data to predict the population of bacteria at a future time?

One way to represent the growth of a population is to use an equation that looks like this:


differential equation example


This is called a differential equation because it shows how a rate of change (in this case, the rate of change of the population) affects one or more other variables. In calculus, a rate of change of a variable is known as a derivative, so a differential equation is one that contains at least one variable and a derivative.

While any equation that contains a derivative and at least one variable is a differential equation, not all such equations are separable. For a differential equation to be separable, the variables must be able to be separated. This means that the equation can be rearranged so that all terms containing one of the variables are on one side of the equal sign, while all terms containing the other variable are on the other side.

For example, the differential equation here is separable because it can be written with all the x variables on one side and all the y variables on the other side, and we end up with:


separable DE


How to Solve an Equation

Now that you know how to recognize a separable differential equation, let's see how to solve one. To solve a separable differential equation, always follow these three steps:

  1. First, separate the variables, just like we did for the equation a moment ago.
  2. Next, integrate each side, as you can see here, where we get:


step 2


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