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.
- 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.
According to Stroud and Booth (2013)* “Solve the following equation: ”
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.
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.
Now I’ll give the derived linear equation a number, say
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!!