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

If you’re looking for more in the first-order ODE, do check-in:

**Solve First-order ODE using transformations**

**Separation of variables in the first-order ODE**

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**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. And I’ll solve the linear ODE in the standard way.

Now I’ll show an example.

**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**

Now here the given differential equation is: And this equation has a term with . So that means it’s a non-linear equation.

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

**Step 1**

As I can see, in this equation is the highest term of . So I’ll divide the equation throughout with .

Hence it will be

Now I’ll simplify it. And that gives

(1)

Now I’ll choose as . Thus it will be

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

So that gives

Now I’ll replace and in equation (2) with and respectively to get

(2)

Thus I’ve converted the non-linear equation to a linear equation. Next, I’ll solve the linear equation.

**Step 2**

Now the equation (2) is comparable to the * standard linear ODE* as

Therefore the integrating factor (I.F.) is

As I can see, in this example, . Thus I.F will be

You can also check out: **Formulas for integrations**

Thus the solution of the differential equation (2) will be

And that means

Now I’ll integrate it to get the solution of the equation (2) as

Here is the integration constant.

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

Now I’ll simplify it. And that gives

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

And this is the answer to the given 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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