Laplace transform of a unit step function. Hi guys, today it’s all about the Laplace transform of the unit step function.

If you’re looking for more in the unit step function, do check-in:

**Heaviside unit step function in Laplace transform**

Want to check out more in Laplace transform of functions?? Here are the links:

*First shift theorem in Laplace transform*

**Laplace transform of functions multiplied by variables**

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

**Laplace transform of a unit step function**

**Solved examples of the Laplace transform of a unit step function**

Disclaimer: 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 (2011)*, “A function is defined by

….determine its Laplace transform.”

**Solution**

Now here I have a step function

First of all, I’ll write it as a unit step function.

If you want to know things in detail, please check out my earlier post on Heaviside unit step function in Laplace transform.

**Step 1**

So this means

Next, I’ll write in term of . And that gives

Then it becomes

(1)

Next, I’ll use the second-shift theorem in Laplace transform to get the Laplace transform of .

**Step 2**

Now from the standard formulas in Laplace transform, I already know that the Laplace transform of the unit step function is . Also, where .

As I can see from equation (1), the first term is . So the Laplace transform of that will be

(2)

And the second term is . So the Laplace transform of that will be

(3)

Then the third term is . So the Laplace transform of that will be

(4)

Now I’ll combine equations (2), (3) and (4) to get the Laplace transform of the function as

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

Now I’ll give you another example.

**Example 2**

According to Stroud and Booth (2011)*, “A function is defined by

Determine .”

**Solution**

Now here I have a step function

First of all, I’ll write it as a unit step function like in example 1.

**Step 1**

So this means

Then I’ll simplify it to get

Next, I’ll write in term of . And that gives

Now I’ll write the other in term of . So it becomes

(5)

Now I’ll get the Laplace transform of this function.

**Step 2**

As I can see from equation (5), the first term is . So the Laplace transform of that will be

(6)

And the second term is . So the Laplace transform of that will be

(7)

Then the third term is . So the Laplace transform of that will be

(8)

In the end, comes the fourth term . So the Laplace transform of that will be

(9)

Now I’ll combine equations (6), (7), (8) and (9) to get the Laplace transform of the function as

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

Now I’ll give you another example.

**Example 3**

According to Stroud and Booth (2011)*, “A function is defined by

Determine (b) the Laplace transform of .”

**Solution**

Now here I have a step function

First of all, I’ll write it as a unit step function like in examples 1 and 2.

**Step 1**

So this means

(10)

Now I’ll get the Laplace transform of this function.

As I can see from equation (10), the first term is . So the Laplace transform of that will be

(11)

And the second term is . So the Laplace transform of that will be

(12)

Then the third term is . So the Laplace transform of that will be

(13)

Now I’ll combine equations (11), (12) and (13) to get the Laplace transform of the function as

Hence I can conclude that 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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