First shift theorem in Laplace transform. Here I’ll talk about the first shift theorem in Laplace transform.

Have a look!!

First shift theorem_compressed

If you’re looking for more in Laplace transform of functions, do check in:

**Laplace transform of functions multiplied by variables**

**Laplace transform of functions divided by a variable**

**Cover up rule in inverse Laplace transform**

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

### First shift theorem in Laplace transform

Suppose the Laplace transform of any function is . This means

Now I multiply the function with an exponential term, say

Then the new function will be

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

Now here comes the first shift theorem of Laplace transform.

It says the Laplace transform of this new function will be

Now I’ll solve some examples of that.

#### Examples of first shift theorem

Disclaimer: None of these examples is 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 are the examples.

##### Example 1

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

##### Solution

Now here I have to find out the Laplace transform of the function

As I have said earlier in this post, the first shift theorem of Laplace transform says if , then .

And in this example, is . And, is .

So that means .

Also, I already know the Laplace transform of from the standard formulas in Laplace transform.

That is

Therefore according to the first shift theorem, will be

Now I’ll simplify it to get the final value of .

Thus it will be

Hence I can conclude that the Laplace transform of the function is .

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 have to find out the Laplace transform of the function

As I have said earlier, the first shift theorem of Laplace transform says if , then .

And in this example, is . And, is .

That means .

I already know the Laplace transform of from the standard formulas in Laplace transform.

That is

Therefore according to the first shift theorem, will be

Now I’ll simplify it to get the final value of .

Thus it will be

Hence I can conclude that the Laplace transform of the function is

And this is the answer to this example.

Now I’ll give you another example.

##### Example 3

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

##### Solution

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

I can also rewrite it as a sum of two functions.

Then it will be

So as a first step, I’ll find out the Laplace transform of the function

Next, I’ll find out the Laplace transform of the function

In the end, I will add these two.

So here goes the first step.

###### Step 1

Now as I have said earlier, the first shift theorem of Laplace transform says if , then .

Thus for the function , is . And, is .

That means .

As per the standard formulas in Laplace transform, the Laplace transform of the function is

Therefore according to the first shift theorem, will be

This means

(1)

Now I’ll go to the second step.

###### Step 2

Here I’ll find out the Laplace transform of the function

Thus for the function , is . And, is .

That means .

As per the standard formulas in Laplace transform, the Laplace transform of the function is

Therefore according to the first shift theorem, will be

(2)

Also I have mentioned earlier that

Therefore I’ll add equations (1) and (2) to get the Laplace transform of the function

Thus it will be

Next, I’ll simplify it to get the final answer.

Hence it will look like

So this gives

Hence I can conclude that the Laplace transform of the function is

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!!

Taiwo says

I really now understand perfectly this post and my engineering mathematics by HK. Dass.

thanks a lot.

Dr. Aspriha Peters says

Thank you. My best wishes for your study.