Laplace transform of t f(t), Laplace transform of functions multiplied by variables. Here I’ll talk about the Laplace transform of functions multiplied by variables, that is, Laplace transform of t f(t). Have a look!!

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

**First shift theorem in Laplace transform**

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

**Cover up rule in inverse Laplace transform**

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

**DOWNLOAD READ & PRINT – Laplace transform of t f(t) | Laplace transform of functions (pdf)**

**Laplace transform of functions multiplied by variables / Laplace transform of t f(t)**

Suppose the Laplace transform of any function is . This means

Now I multiply 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

Now the Laplace transform of this new function will be

Now I’ll solve some examples of that.

**Examples of Laplace transform of functions multiplied by variables / Laplace transform of t f(t)**

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**

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

Now as I have said earlier in this post, if , then .

In this example, is .

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

That is

If I simplify it, then it will be

Now as per the formula of the Laplace transform of functions multiplied by variables, will be

Therefore, I’ll differentiate with respect to .

For that, I’ll use the standard formulas in differentiation.

So I get

Next, I use the formula

That 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 the Laplace transform of the function

I can also rewrite the function as

Also, I know that .

Therefore I can say the Laplace transform of the function will be .

So, as a first step, I’ll get the Laplace transform of the function .

**Step 1**

Now from the standard formulas in Laplace transform, I can say that

Thus the Laplace transform of the function will be

Next I’ll differentiate with respect to .

For that, I’ll use the quotient rule of differentiation.

That gives

Now I’ll simplify it.

So it becomes

Thus I can say

As a next step, I’ll determine

**Step 2**

Thus will be

Next I’ll differentiate with respect to .

For that, again I’ll use the quotient rule of differentiation.

That gives

Now I’ll simplify it.

So it becomes

Next I cancel from both top and bottom, that is the numerator and the denominator. This is because

Therefore it will be

Now I’ll simplify it a bit more to get

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

This is the answer to this 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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