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

If you are looking for more in Laplace transform of functions, do check-in:

**First shift theorem in Laplace transform**

**Laplace transform of functions multiplied by variables**

**Cover up rule in inverse Laplace transform**

**How to use partial fractions in inverse Laplace transform**

**DOWNLOAD READ & PRINT – Laplace transform of a function divided by variables (pdf)**

**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

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**

Now here I will find out Laplace transform of the function .

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

**Step 1**

So I can say

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

Therefore it will be

Next, I’ll substitute in the limit to get

So this gives

Thus I can say that this limit exists.

Now I will find out Laplace transform of the function .

**Step 2**

Also, from the formulas in Laplace transform, I already know that

Therefore the Laplace transform of the function will be

Now I’ll integrate the right-hand side. And this gives

Since , I can say that

Next, I’ll substitute the limits. As I can see, here the higher limit is and the lower limit is . Thus the Laplace transform of the function will be

When we add any number to the infinity , the result is always infinity . So in this example,

Also, . Therefore I can say that,

Hence I can conclude that the Laplace transform of the function will be

And 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 **

Now here I will find out Laplace transform of the function .

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

**Step 1**

So I can say

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

Therefore it will be

Next, I’ll substitute in the limit to get

So this gives

Thus I can say that this limit exists.

Now I will find out Laplace transform of the function .

**Step 2**

As per the formulas in Laplace transform, I already know that

Therefore the Laplace transform of the function will be

Now I’ll integrate the right-hand side. And this gives

Next, I’ll substitute the limits. As I can see, here the higher limit is and the lower limit is . Thus the Laplace transform of the function will be

When we subtract any number from the infinity , the result is always infinity . Also, when we add any number to , the result is always .

So in this example,

Also, . Therefore I can say that,

Hence I can conclude that the Laplace transform of the function will be

And this is the answer to 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!!

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