First order linear ODE in first-order ordinary differential equations. Dear friends, today I will talk about the first-order linear ODE in first-order ordinary differential equations. What is it and how to solve it?

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

**Solve First-order ODE using transformations**

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

**First-order linear ODE in first-order ordinary differential equations – 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

As I have already mentioned in the **formulas for functions**, , I can say that

Next, I will substitute this value of in equation (1). Then I get

Also, I already know that . So I can say that

**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). Then I get

Since , I can say that . So this gives

Next, I’ll integrate it. Then I get

Now is the integration constant here. If I simplify it, I get

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 substitute this value in equation (1). Then I get the integrating factor as

Now I’ll simplify it. So I 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). Then I get

Next, I’ll integrate it. So it gives

Now is the integration constant here. Then I’ll simplify it. So that means

(3)

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’ll substitute to the general solution of the given first-order ODE, that is, equation (3). So that gives

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

So I can say that which means .

Now I’ll put back in equation (3). So that gives

Thus I can say that is the solution of the given equation for . Hence I can conclude that this is the solution to the given example.

Dear friends, this is the end of today’s post on the First-order linear ODE in first-order ordinary differential equations. 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!!

###### *Reference: K. A. Stroud and Dexter J. Booth (2013): Engineering Mathematics, Industrial Press, Inc.; 7th edition (March 8, 2013), Chapter: First-order Differential Equations, Further problems 25, p. 1000, Q. No.s 44 (Example 1), 34 (Example 2).

**DOWNLOAD, READ & PRINT – First order linear ODE (pdf)**

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.

ClassCrown says

Thanks for this!

Dr. Aspriha Peters says

Thanks a lot.