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

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**

First-order linear ODE_compressed

**First-order linear ODE – how to identify it?**

Ok, so it’s not a difficult thing to identify a first-order linear ODE. Any linear first-order differential equation has a general form

And it will be non-linear if there will be any term with

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

**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 of the 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**

Now here the given equation is:

As you can see, I can compare it with the standard form of a linear first order ODE.

Also I have already mentioned it earlier that the standard form of a linear first order ODE is

Therefore I can say that in this example,

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

**Step 1**

First of all, I will integrate

So this gives

Next I will substitue this value of

So I can say

**Step 2**

Now I will get the solution of the differential equation. Thus I will substitute the values of the integrating factor (IF) and

So this gives

Next, I’ll integrate it to get

Also

Now I will give another example.

**Example 2**

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

**Solution**

Now here the given equation is:

Next, I’ll compare it with the standard form of a linear first order ODE. As you know already from example 1, the standard form of a linear first order ODE is

Therefore I can say that in this example

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

**Step 1**

First of all, I will integrate

So I can say that

Next, I will substitue this value in equation (1) to get the integrating factor as

Then I’ll simplify it to get

**Step 2**

Now I’ll get the solution of the differential equation. So I’ll substitute the values of the integrating factor (IF) and

Then I’ll integrate it to get

And here

**Step 3**

Next, I substitute

Then I’ll simplify it to get the value of

Thus

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!!

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

Your academic insight is quite helpful.

ibrahim R.

Hi Ibrahim,

Thank you very much.