How Do You Find the Laplace Inverse of \( \frac{40}{(s^2+4s+5)^2} \)?

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Homework Help Overview

The discussion revolves around finding the Laplace inverse of the expression \( \frac{40}{(s^2+4s+5)^2} \). The subject area pertains to Laplace transforms and their inverses, particularly focusing on techniques for handling complex expressions in the context of differential equations.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss completing the square in the denominator and the implications for the Laplace inverse, noting the expected involvement of sine and cosine functions. Some express uncertainty about the path to the solution, while others suggest using partial fractions or convolution as alternative methods.

Discussion Status

There is an ongoing exploration of different methods to approach the problem, with some participants providing insights into potential techniques like convolution and partial fractions. However, no consensus has been reached on a specific method or solution.

Contextual Notes

Participants mention a reluctance to use complex numbers, indicating a preference for real-number approaches, which may affect the methods discussed. There is also a reference to using software tools for assistance, highlighting varying levels of familiarity with the topic.

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Homework Statement



laplace inverse of (40/(s^2+4s+5)^2)?

Homework Equations



I completed the square in the denominator to get 40/((s+2)^2+1)^2
I know that I will get cosines and sines from the shape of it in the laplace inverse; however I'm stuck.


The Attempt at a Solution

 
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I get the follwing result:
[itex]20e^{-2t}(\sin(t)-t\cos(t))[/itex]
But couldn't find any formula, so i did it with Maple 15
 
gethelpelectr said:

Homework Statement



laplace inverse of (40/(s^2+4s+5)^2)?

Homework Equations



I completed the square in the denominator to get 40/((s+2)^2+1)^2
I know that I will get cosines and sines from the shape of it in the laplace inverse; however I'm stuck.


The Attempt at a Solution


It will be a lot easier if you first factor the expression p = s^2 + 4s + 5, then convert your expression 1/p^2 to partial fractions.

RGV
 
dikmikkel This answer is correct; however I don't know how to get there.
Ray Vickson, we are not used to getting complex numbers.
 
gethelpelectr said:
dikmikkel This answer is correct; however I don't know how to get there.
Ray Vickson, we are not used to getting complex numbers.

You should get used to it; they are part of a standard toolkit and are used routinely in Physics, Engineering and Applied Math. However, if you don't want to use complex quantities you can use the convolution theorem instead: first get the inverse Laplace transform of 1/(s^2 + 4s + 5), then find that of 1/(s^2 + 4s + 5)^2 by convolution.

RGV
 
Last edited:

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