First order linear ODE. Dear friends, today I will talk about first order linear ODE. What is it and how to solve it?

#### First order linear ODE – how to identify it?

It’s not a difficult thing to identify a first-order linear ODE.

Any linear first order differential equation has a general form

It will be non-linear if there will be any term with something like that.

Now the next part is ‘how to solve a first-order linear ODE’?

If interested, you can also check out more posts in first order ODE such as

- First-order homogeneous ODE
- Solve First-order ODE using transformations
- Separation of variables in the first-order ODE
- Bernoulli’s equations

#### First order linear ODE – how to solve it?

I have already said that it has a standard form of

Now I can solve it in two steps.

First, I have to get the integrating factor. In short form, it is IF.

Then to solve the differential equation with the standard formula.

(1)

The solution of the equation is:

(2)

Now I will solve some examples.

#### Examples on first order linear ODE

Note: 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 first example.

##### Example 1

According to Stroud and Booth (2013)* “Determine the general solution of the equation ”

##### Solution

Here the given equation is:

Now I can compare it with the standard form of a linear first order ODE.

I have already mentioned it earlier as

Therefore I can say that

###### Step 1

Now the first step will be to get the integrating factor.

For that, first of all, I will integrate . Thus it will be

Next I will substitue this value in equation (1) to get

###### Step 2

Now I will get the solution of the differential equation.

Thus I will substitute all the values in equation (2) to get

Here is the integration constant.

This is the solution to the given equation.

Now I will go to the next example.

##### Example 2

According to Stroud and Booth (2013)* “Solve the equation given that when ”

##### Solution

Here the given equation is:

Now I will divide it through out with to get

Now I can compare it with the standard form of a linear first order ODE.

I have already mentioned it earlier as

Therefore I can say that

###### Step 1

Now the first step will be to get the integrating factor.

For that, first of all, I will integrate . Thus it will be

Next I will substitue this value in equation (1) to get

###### Step 2

Now I will get the solution of the differential equation.

Thus I will substitute all the values in equation (2) to get

Here is the integration constant.

This is the general solution of the given equation. Now I have to find out the particular solution of the equation for

###### Step 3

Thus I substitute in the general solution of the equation to get

Thus is the solution of the given equation for

Dear friends, this is the end of my 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!!

ibrahim R. says

Thank you very much for your effort with your academic help.

Your academic insight is quite helpful.

ibrahim R.

Dr. Aspriha Peters says

Hi Ibrahim,

Thank you very much.