Finding Inverse Laplace Transform with Fractional Powers and Convolution

In summary, the conversation discusses finding the inverse Laplace transform of 1/(s-2)^3 + 25/(s+1)(s-2)^2 + s/(s-2)^2. The first two terms are simplified to (1/2)e^(2t)t^2 + (25/7)e^(-t)-(25/7)e^(2t)+(75/7)te^(2t), while the third term is simplified to e^(2t) + 2e^(2t)t. The conversation also mentions using convolution and suggests writing the numerator in terms of (s-2).
  • #1
Sol-chan
6
0
Homework Statement

Find the inverse laplace transform of:

1/(s-2)3 + 25/(s+1)(s-2)2 + s/(s-2)2


The attempt at a solution

I get (1/2)e2tt2 + (25/7)e-t-(25/7)e2t+(75/7)te2t for the first two, but I'm not even sure where to start for s/(s-2)2. I was thinking it might use convolution, but I'm not sure that I understand how convolution works... I'd really just like a hint to get me started on it...
 
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  • #2
1/(s-2)2 looks like a shifted form of 1/s2, so you want to write numerator in terms of (s-2), i.e. s = (s-2)+2.
 
  • #3
Okay, so then I would have (s-2)/(s-2)2 + 2/(s-2)2

which is 1/(s-2) + 2/(s-2)2

so then I'd get e2t + 2e2tt
 
  • #4
That's right. You may want to check your algebra on your earlier work. Mathematica gets a different answer than you do.
 

1. What is an Inverse Laplace Transform?

The Inverse Laplace Transform is a mathematical operation that takes a function in the complex frequency domain and converts it back into a function in the time domain. It is used in engineering, physics, and other scientific fields to solve differential equations and analyze systems.

2. How is the Inverse Laplace Transform calculated?

The Inverse Laplace Transform is calculated using a table of known Laplace transforms and the inverse transform formula, which involves integrating the function multiplied by an exponential term.

3. What are the applications of the Inverse Laplace Transform?

The Inverse Laplace Transform has many applications, including solving differential equations, analyzing linear systems, and modeling physical systems such as electrical circuits, chemical reactions, and mechanical systems.

4. Are there any limitations to the Inverse Laplace Transform?

Yes, there are limitations to the Inverse Laplace Transform. It can only be used for functions that have a Laplace transform, meaning they are defined for all positive real numbers and have a finite number of discontinuities. It also cannot be used for functions with complex poles or singularities.

5. How is the Inverse Laplace Transform related to the Laplace Transform?

The Inverse Laplace Transform is the reverse operation of the Laplace Transform. It converts a function from the complex frequency domain back into the time domain, while the Laplace Transform converts a function from the time domain to the complex frequency domain. Together, they are powerful tools in solving differential equations and analyzing systems.

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