Laplace Transform L[x(t)] given, find L[tx(t)]

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Discussion Overview

The discussion revolves around finding the Laplace transform of the function \( tx(t) \) given the Laplace transform of \( x(t) \). Participants explore the implications of the problem statement and various approaches to solving it, including the application of properties of Laplace transforms.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants express uncertainty about the specific requirements of the problem, questioning whether it is asking for the Laplace transform of \( t x(t) \).
  • One participant suggests that the property \( L[tf(t)] = -\frac{dF(s)}{ds} \) is relevant to the solution.
  • Another participant provides the derivative of the given Laplace transform \( L[x(t)] = \frac{s+4}{s^2 + 1} \) as part of their approach to finding \( L[tx(t)] \).
  • There is a discussion about whether simplifying the derivative is necessary to obtain the final form of \( L[tx(t)] \).
  • One participant acknowledges a mistake regarding the negative sign in the derivative, indicating the need for correction in their calculations.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the interpretation of the problem or the steps required to solve it. There are multiple viewpoints on how to approach the solution, and some uncertainty remains regarding the application of the derivative property.

Contextual Notes

Some participants request explicit attempts at solutions, indicating that the problem may require detailed mathematical steps that have not been fully articulated. The discussion reflects varying levels of understanding and confidence in applying the Laplace transform properties.

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Homework Statement



If L[x(t)] = (s + 4)/(s2 + 1), find L[tx(t)]

Homework Equations


Laplace transform:

F(s) = 0∫ f(t)e-stdtLaplace table

The Attempt at a Solution


Clearly it's not just asking for a Laplace transform. Not sure what it's specifically asking to be honest.

t multiplied by whatever is inside the equation definitely isn't the answer.
 
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Color_of_Cyan said:

Homework Statement



If L[x(t)] = (s + 4)/(s2 + 1), find L[tx(t)]

Homework Equations


Laplace transform:

F(s) = 0∫ f(t)e-stdtLaplace table

The Attempt at a Solution


Clearly it's not just asking for a Laplace transform.
Clearly it is. There are a couple of approaches you could take, but I'd like to see what you have tried before I share them with you.
Color_of_Cyan said:
Not sure what it's specifically asking to be honest.

t multiplied by whatever is inside the equation definitely isn't the answer.
I don't know what this means...
 
Color_of_Cyan said:
Not sure what it's specifically asking to be honest.

Is it asking for the Laplace transform of the function ##t x(t)## (i.e., the function obtained by multiplying ##x(t)## by ##t##), given the Laplace transform of ##x(t)##? That's what it looks like to me, but I'd like to make sure you have transcribed the problem statement correctly.

Also, please show us explicitly your attempt at a solution(i.e., with equations showing the steps of the calculation you attempted). Just describing it in words isn't enough.
 
Okay, there seems to be one property that sticks out for this:

L[tf(t)] = -dF(s)/ds

L[x(t)] = (s+4)/(s2 + 1)

L[f(t)] = (s+4)/(s2 + 1). Then find L[tf(t)]

$$\frac{d} {ds} [\frac {s+4} {s^2 + 1}] $$

Just this derivative?

= (d/ds)[(s+4)/(s2 + 1)-1]

=
$$ \frac {(1)(s^2 + 1) - (2s)(s+4)} {{(s^2 + 1)}^2} $$

Would I need only simplify the rest of this to get L[tx(t)] ?
 
Color_of_Cyan said:
Okay, there seems to be one property that sticks out for this:

L[tf(t)] = -dF(s)/ds

L[x(t)] = (s+4)/(s2 + 1)

L[f(t)] = (s+4)/(s2 + 1). Then find L[tf(t)]

$$\frac{d} {ds} [\frac {s+4} {s^2 + 1}] $$

Just this derivative?

= (d/ds)[(s+4)/(s2 + 1)-1]

=
$$ \frac {(1)(s^2 + 1) - (2s)(s+4)} {{(s^2 + 1)}^2} $$

Would I need only simplify the rest of this to get L[tx(t)] ?
Yes
 
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Okay, thanks! :)

Edit: Oh wait I forgot that there was a negative sign by the derivative, so

(s2 + 8s - 1)/(s2 + 1)2
 
Last edited:

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