Laplace transform of a function divided by variables. Today it’s about Laplace transform of a function divided by variables.

### Laplace transform of a function divided by variables

Suppose the Laplace transform of any function is . This means

Now I divide the function with a variable, say

Then the new function will be

Next, I want to find out the Laplace transform of the new function

For that, first of all, I have to check a limit.

And that is

If this limit exists, then only it will be possible to get the Laplace transform of the function

Suppose this limit exists.

Then I’ll find out the Laplace transform of the function .

Let me choose it as, say

Then the Laplace transform of the function will be

I have also written a few more posts on Laplace transform like

- First shift theorem in Laplace transform
- Laplace transform of a function multiplied by a variable
- Formulas in Laplace transform

Now I’ll solve some examples of that.

#### Examples of Laplace transform of a function divided by variables

Disclaimer: These examples do not belong to me. I have chosen these from a book. At the end of the post, I have given the due reference.

##### Example 1

According to Stroud and Booth (2011)* “Determine the Laplace transform of the following function: .”

##### Solution

Here I have to find out Laplace transform of the function .

First of all, I’ll check the limit of this function.

So I can say

Then I’ll use l Hôpital’s rule to find out the limit of this function.

Therefore it will be

Thus I can say that this limit exists.

Now I will find out Laplace transform of the function .

From the formulas in Laplace transform, I already know that

Therefore the Laplace transform of the function will be

This gives

This is the answer to this example.

Now I’ll give you another example.

##### Example 2

According to Stroud and Booth (2011)* “Determine the Laplace transform of the following function: .”

##### Solution

Here I have to find out Laplace transform of the function .

First of all, I’ll check the limit of this function.

So I can say

Then I’ll use l Hôpital’s rule to find out the limit of this function.

Therefore it will be

Thus I can say that this limit exists.

Now I will find out Laplace transform of the function .

From the formulas in Laplace transform, I already know that

Therefore the Laplace transform of the function will be

This gives

Hence I can conclude that this is the answer to this example.

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!!

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