Second shift theorem. Hello friends, today it’s about the second shift theorem in inverse Laplace transform.

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

**Heaviside unit step function in Laplace transform**

**Laplace transform of the unit step function**

Want to know about the first shift theorem, click at the link below:

**First shift theorem in Laplace transform**

**Second shift theorem**

Now here I’ll show how to use the second shift theorem in inverse Laplace transform. I have already shown how to use it in the Laplace transform of a unit step function. (check the link above!)

The second shift theorem in Laplace transform says

where .

**Examples of second shift theorem in inverse Laplace transform**

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)*, “Determine the function whose transform is

”

**Solution**

Now here the Laplace transform of the given function is

Next, I’ll simplify it to get

As I can see from the second shift theorem, corresponds to . Also, corresponds to . Therefore the inverse Lalplace transform of is

So the function is .

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)*, “If , determine .”

**Solution**

Now here the function is

So that means the Laplace transform of the function is

Next, I’ll simplify it to get

So I can say,

Now I’ll evaluate it. As in example 1, corresponds to , corresponds to and corresponds to .

Thus, as per the formulas in second shift theorem, the function will be

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)*, “If , determine .”

**Solution**

Now in this example, the Laplace transform of the given function is

Next, I’ll simplify it to get

As in examples 1 and 2, corresponds to and corresponds to .

Thus the function will be

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

Now I’ll give you another example.

**Example 4**

According to Stroud and Booth (2011)*, “If , find in terms of the unit step function.”

**Solution**

Now here the function is

So that means the Laplace transform of the function is

Next, I’ll simplify it to get

So I can say,

Now I’ll evaluate it. As in example 1, corresponds to , corresponds to and corresponds to .

Thus, as per the formulas in second shift theorem, the function will be

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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