Cover up rule in inverse Laplace transform. Hello friends, today it’s all about the cover up rule in inverse Laplace transform.

Have a look!!

### Cover up rule in inverse Laplace transform

Cover up rule in inverse Laplace transform is quite a straightforward thing.

Suppose I have to find the inverse Laplace transform of or .

In both of these examples, one thing is common. That is the denominator.

In these examples, the denominator has two or three factors. But each factor has only , not or .

So in this kind of situation, I can use the cover up rule.

First of all, I’ll write the expression in a partial fraction form. So for the first example, it will be

Now instead of solving it in the usual partial fraction style, I’ll use the cover up rule.

So this means the coefficient of , that is the value of , will be

Similarly, the value of will be

Now I’ll give some examples of the cover up rule.

#### Examples of the cover up rule in inverse Laplace transform

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.

So here are the examples.

##### Example 1

According to Stroud and Booth (2011) “Determine the inverse transformation of the following: ”

##### Solution

Here I have to find the inverse Laplace transform of

Now the denominator has two factors. And both of them has only , not or any other higher power of .

So I can rewrite this expression in partial fraction form as

Now I’ll use the cover up rule to get the values of and .

So the value of is the coefficient of .

Thus it will be

In the same way, now I’ll get the value of .

Thus it will be

Therefore the expression becomes

Now I’ll use formulas in Laplace transform to get the inverse transformation of this expression.

Thus it will be

So this gives

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

Now I’ll go to the next example.

##### Example 2

According to Stroud and Booth (2011) “Determine the inverse transformation of the following: ”

##### Solution

Here I have to find the inverse Laplace transform of

Now the denominator has two factors. And both of them has only , not or any other higher power of .

So I can rewrite this expression in partial fraction form as

Now I’ll use the cover up rule to get the values of and .

So the value of is the coefficient of .

Thus it will be

In the same way, now I’ll get the value of .

Thus it will be

Therefore the expression becomes

Now I’ll use formulas in Laplace transform to get the inverse transformation of this expression.

Thus it will be

So this gives

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

But there are many other cases where I’ll use the techniques of partial fractions first and then get the inverse Laplace transformation of them. Soon I’ll write about those things as well.

Dear friends, this is the end of my 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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