What is the derivation of 2e^{2t}\sin{t} in Laplace transform?

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Homework Help Overview

The discussion revolves around the derivation of the inverse Laplace transform of a specific function, particularly focusing on the term involving \(2e^{2t}\sin{t}\). Participants are examining the steps taken to arrive at the correct expression and clarifying the components involved in the transformation process.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to understand the derivation of a specific term in the inverse Laplace transform and questions the discrepancy between their calculation and the book's answer. Other participants discuss the necessity of adjusting the numerator in the canonical form to account for the additional term involving sine.

Discussion Status

Participants are actively engaging with the problem, with some providing insights into the necessary adjustments for the inverse transform. There is recognition of differing interpretations regarding the correct form of the function, and the discussion is ongoing without a clear consensus on the resolution.

Contextual Notes

One participant notes their recent challenges in keeping up with the coursework due to personal circumstances, which may influence their understanding of the material.

faust9
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OK, here's the problem

[tex]L^{-1}\{\frac{s}{s^2+4s+5}\}[/tex]

What I did:
[tex]L^{-1}\{\frac{s}{(s+2)^2+1}\}[/tex]

[tex]=e^{2t}\cos{t}[/tex]

The book says:
[tex]=e^{2t}\cos{t}-2e^{2t}\sin{t}[/tex]


Where did the [itex]2e^{2t}\sin{t}[/itex] come from?

Mahalo.
 
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Mathematica is saying that it's neither of those...

cookiemonster
 
The cannonical form has an (s+2) in the numerator, not just s (so your invesrse transform calculation was actually incorrect). In order to account for that, you have to add another rational function that has a numerator of -2, which just happens to be (-2) times the cannonical form with sin instead of cos.
 
Thanks a lot. I missed about a month of Diff Eq due to illness and an auto accident so I've been learning it all on my own. I appreciate the Q&A forum here and all those who help out.
 

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