Inverse laplace transform question

In summary, to find the inverse Laplace transform of s/(s-2)^2, we can use the "shift" property to rewrite it as (1+t)e^{2t}. Then, using partial fractions, we can find the inverse transform to be [2te^{2t}+e^{2t}] u(t), without needing to use the residue theorem.
  • #1
cabellos
77
1
How do i find the inverse laplace transform of s/(s-2)^2
 
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  • #2
take the transform of [tex] f(t)=(1+t)e^{2t} [/tex]

using the "shift" property so..

[tex] F(s+a)=L[e^{-at}f(t)] [/tex] using

[tex] \frac{s+2}{s^{2}}=s^{-1}+2s^{-2} [/tex]

the inverse of above is just 1+t then multiply it by exp(2t) and you get it without recalling "residue theorem"...:tongue2:
 
  • #3
Use partial fractions:

[tex]\frac{s}{(s-2)^2} = \frac{A}{(s-2)^2} + \frac{B}{s-2}[/tex]

To find A, multiply both sides by [tex]s-2[/tex] and evaluate at [tex]s=2[/tex]:

[tex]s = A[/tex]

[tex]A = 2[/tex]

Now to find B, go back to the original expression again, and multiply (again) both sides by [tex]s-2[/tex],
then differentiate with respect to s and evaluate at [tex]s=2[/tex]:

[tex]\frac{d}{ds}s = \frac{d}{ds}[B(s-2)][/tex]

[tex]B = 1[/tex]

Now plug A and B into the original expression:

[tex]\frac{2}{(s-2)^2} + \frac{1}{s-2}[/tex]

So the inverse Laplace would be:

[tex]L^{-1} = [2te^{2t}+e^{2t}] u(t)[/tex]
 

1. What is the inverse Laplace transform?

The inverse Laplace transform is a mathematical operation that takes a function in the complex frequency domain and converts it into a function in the time domain. It is the reverse process of the Laplace transform, which transforms a function in the time domain into the complex frequency domain.

2. How is the inverse Laplace transform calculated?

The inverse Laplace transform is calculated using a table of known Laplace transform pairs or through the use of integral calculus. The most common method is using the table, which matches the function in the complex frequency domain to its corresponding function in the time domain.

3. What is the significance of the inverse Laplace transform in scientific research?

The inverse Laplace transform is a useful tool in various fields of science and engineering, including signal processing, control systems, and quantum mechanics. It allows scientists to analyze and understand the behavior of systems in the time domain by studying their representation in the complex frequency domain.

4. Can the inverse Laplace transform be applied to any function?

No, the inverse Laplace transform can only be applied to functions that are Laplace transformable, meaning they have a valid Laplace transform. Some functions, such as those with infinite discontinuities, do not have a Laplace transform and therefore cannot have an inverse Laplace transform.

5. Are there any limitations to the inverse Laplace transform?

Yes, there are limitations to the inverse Laplace transform. One limitation is that it cannot be used to find the inverse of a function with multiple roots or poles that are too close together. Additionally, the inverse Laplace transform may not exist for functions with certain types of singularities, such as essential singularities.

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