Differential Equations - Separable Variables

If you want to describe the world around you, be it the forces acting on a body, the growth of a virus or the temperature of the coffee in your cup, you will be dealing with differential equations. On this page we will look at the simplest type: differential equations in which we can separate the variables. To be successful with the topic on this page, you will need to have strong foundations in the following areas: integration techniques, manipulation of logarithms and exponents and partial fractions. It is recommended that you refresh your techniques in those areas if you are not feeling 100% confident.


Key Concepts

On this page, you should learn to

  • recognise differential equations with separable variables - in the form \(\frac{dy}{dx}=f(x)g(y)\)
  • solve these differential equations using integration

Essentials

The following videos will help you understand all the concepts from this page

Example 1 - Finding a General Solution

In the following video we look at how we can solve separable differential equations

Notes from the video

 

Example 2 - Finding a Specific Solution

In the following video, we will see an example of finding a particular solution to a differential equation. The variables in this differential equation are separable

Notes from the video

 

 

Summary

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Test Yourself

To solve differential equations, we are often required to manipulate exponents and logarithms. Here is a quiz that will give you some practice in what you need


START QUIZ!

When giving the general solution to differential equation, we have to manipulate arbitrary constants. Try this quiz to ensure that you know how to do this


START QUIZ!

Here is a quiz that gets you to practise solving variables separable differential equations


START QUIZ!

Exam-style Questions

Question 1

Solve the differential equation given that y(0) = 2

\({\large \frac{dy}{dx}=\frac{e^x}{y}, \quad y>0}\)

Hint

Full Solution

 

Question 2

a) Write \({\huge \frac{1}{4-x^2}}\)as the sum of two partial fractions

b) Hence, given that y(0) = 0, find the particular solution of the differential equation in the form

Hint

Full Solution

 

Question 3

a) Express \(\frac{3}{x(x-3)}\) as the sum of two partial fractions

The population of a species of fish can be modelled by the differential equation \(\large \frac{\text{d}N}{\text{d}t}=\frac{2}{3}N(N-3)cos2t\)

where N = population in thousands, t = time in years

b) Given that initially the population of fish is 4000, show that \(\large N=\frac{12}{4-e^{sin{2t}}}\)

c) How many days during the first year is the population of fish above 8000?

Hint

Full Solution

 

MY PROGRESS

How much of Differential Equations - Separable Variables have you understood?