Bernoulli’s equations, non-linear equations in an ODE. Dear friends, today I’ll talk about Bernoulli’s equations in an ODE.

### Bernoulli’s equations, non-linear equations in ODE

###### What are Bernoulli’s equations?

Any first order ordinary differential equation (ODE) is linear if it has terms only in .

But if the equation also contains the term with a higher degree of , say, or more, then it’s a non-linear ODE.

The standard form of a linear ODE is

Similarly the standard form of a non-linear ODE is

First order non-linear ODE of this form is also known as Bernoulli’s equation.

###### How can I solve Bernoulli’s equations?

Now, in order to solve a Bernoulli’s equation, first of all, I’ll convert it to a linear equation.

For that,

- I’ll divide the equation with .
- Next, I have to choose as .
- Then I’ll rewrite the equation in terms of

In the end, the non-linear equation will be transferred to a linear one.

Thus I can solve the linear ODE in the standard way.

If interested, you can also check out further posts on the first-order ODE such as

- First-order linear ODE
- Solve First-order ODE using transformations
- Separation of variables in the first-order ODE
- First-order homogeneous ODE

#### Example of the Bernoulli’s equations in ODE

Disclaimer: None of these examples are mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.

So here is the example.

##### Example

According to Stroud and Booth (2013)* “Solve the following equation: ”

##### Solution

Here the given differential equation is:

Now this equation has a term with

That means it’s a non-linear equation.

So my first task is to convert this equation to a linear one.

###### Step 1

In this equation is the highest term of .

Thus, first of all, I’ll divide the equation throughout with .

So it will be

Now I’ll simplify it to get

Now I’ll choose as .

Thus it will be

Next, I’ll differentiate both sides with respect to to get

Now I’ll replace and with and respectively to get

Thus I’ve managed to convert the non-linear equation to a linear equation

My next task is to solve the linear equation.

###### Step 2

Now I’ll give the derived linear equation a number, say

(1)

This is comparable to the standard linear ODE as

Therefore the integrating factor (I.F.) is

You can also check out: Formulas for integrations

Thus the solution of the differential equation (1) will be

Now I’ll substitute the I.F. and integrate to get the solution of the equation (1) as

Here is the integration constant.

Next I’ll bring back in the answer. Thus it will be

Now I’ll simplify it a bit more like

Hence I can conclude that the general solution of the equation is

This is the answer to this example.

Dear friends, this is the end of today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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