First order linear ODE. Dear friends, today I will talk about the 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 something like that.

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 . Thus it will be

So this gives

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

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 in equation (2) to get

So this gives

Next, I’ll integrate it to get

Also is the integration constant here. Hence I can conclude that is the solution to the given equation.

Now I will give another example.

**Example 2**

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

**Solution**

Now here the given equation is: . First of all, I will divide it through out with to get

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 . Thus it will be

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 in equation (2) to get

Then I’ll integrate it to get

And here is the integration constant. Thus is the general solution of the given equation. Now I have to find out the particular solution of the equation for

**Step 3**

Next, I substitute in the general solution of the equation to get

Then I’ll simplify it to get the value of . And that will be

Thus is the solution of the given equation for . Hence I can conclude that this is the solution of 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!!

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.