Correcting a Laplace Transform Problem

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Homework Help Overview

The discussion revolves around a Laplace Transform problem, specifically addressing the integration of the function involving \( e^{-st} \) and \( \cos(at) \). Participants are examining the correct application of integration techniques and limits in the context of Laplace Transforms.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the integration by parts method applied to the Laplace Transform, questioning the correctness of their results and the presence of terms involving \( t \) in the final answer. Some express confusion regarding the limits of integration and the implications of integrating out \( t \).

Discussion Status

There is an ongoing examination of the integration process, with one participant noting a sign error that affected their results. Others are providing insights on the necessity of applying limits correctly, and some participants are reflecting on the implications of their findings after correcting their approach.

Contextual Notes

Participants are working under the constraints of homework guidelines, which may limit the information they can reference or the methods they can use. The discussion highlights the importance of understanding the integration limits in the context of Laplace Transforms.

jayanthd
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Homework Statement



Why I am getting wrong answer related to this Laplace Transforms problem?

According to the book "Higher Engineering Mathematics 6th edition by John O Bird" page no. 583 one should get

(e[itex]^{-st}[/itex]/(s[itex]^{2}[/itex] + a[itex]^{2}[/itex]))(a sin at - s cos at)

Homework Equations



∫e[itex]^{-st}[/itex]cos at dt

The Attempt at a Solution

u = e[itex]^{-st}[/itex]

du = -se[itex]^{-st}[/itex] dt

Let dv = cos at dt

v = (sin at)/ a

Integrating by parts

∫e[itex]^{-st}[/itex]cos at dt =

(e[itex]^{-st}[/itex] sin at / a) + (s/a)∫e[itex]^{-st}[/itex]sin at dt

= (e[itex]^{-st}[/itex] sin at / a) + (s/a)[(-e[itex]^{-st}[/itex] cos at / a) - (s/a)∫e[itex]^{-st}[/itex]cos at dt]

= (e[itex]^{-st}[/itex] sin at / a) - (s/a[itex]^{2}[/itex] )(e[itex]^{-st}[/itex] cos at) - s[itex]^{2}[/itex]/a[itex]^{2}[/itex])∫e[itex]^{-st}[/itex]cos at dt]

Rearranging

(1 + (s[itex]^{2}[/itex]/a[itex]^{2}[/itex]))∫e[itex]^{-st}[/itex]cos at dt =

(e[itex]^{-st}[/itex] sin at / a) - (s/a[itex]^{2}[/itex])(e[itex]^{-st}[/itex] cos at)

= (e[itex]^{-st}[/itex]/a[itex]^{2}[/itex])(a sin at - s cos at)

∫e[itex]^{-st}[/itex]cos at dt =

(a[itex]^{2}[/itex]/(a[itex]^{2}[/itex] + s[itex]^{2}[/itex]))((e[itex]^{-st}[/itex]/a[itex]^{2}[/itex])(a sin at - s cos at))

= ((e[itex]^{-st}[/itex]/ (s[itex]^{2}[/itex] + a[itex]^{2}[/itex]))(a sin at - s cos at)
 
Last edited:
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jayanthd said:

Homework Statement



Why I am getting wrong answer related to this Laplace Transforms problem?

According to the book "Higher Engineering Mathematics 6th edition by John O Bird" page no. 583 one should get

(e[itex]^{-st}[/itex]/(s[itex]^{2}[/itex] + a[itex]^{2}[/itex]))(a sin at - s cos at)


Homework Equations



∫e[itex]^{-st}[/itex]cos at dt

The Attempt at a Solution




u = e[itex]^{-st}[/itex]

du = -se[itex]^{-st}[/itex] dt

Let dv = cos at dt

v = (sin at)/ a

Integrating by parts

∫e[itex]^{-st}[/itex]cos at dt =

(e[itex]^{-st}[/itex] sin at / a) + (s/a)∫e[itex]^{-st}[/itex]sin at dt

= (e[itex]^{-st}[/itex] sin at / a) + (s/a)[(-e[itex]^{-st}[/itex] cos at / a) - (s/a)∫e[itex]^{-st}[/itex]cos at dt]

= (e[itex]^{-st}[/itex] sin at / a) - (s/a[itex]^{2}[/itex] )(e[itex]^{-st}[/itex] cos at) - s[itex]^{2}[/itex]/a[itex]^{2}[/itex])∫e[itex]^{-st}[/itex]cos at dt]

Rearranging

(1 + (s[itex]^{2}[/itex]/a[itex]^{2}[/itex]))∫e[itex]^{-st}[/itex]cos at dt =

(e[itex]^{-st}[/itex] sin at / a) - (s/a[itex]^{2}[/itex])(e[itex]^{-st}[/itex] cos at)

= (e[itex]^{-st}[/itex]/a[itex]^{2}[/itex])(a sin at - s cos at)

∫e[itex]^{-st}[/itex]cos at dt =

(a[itex]^{2}[/itex]/(a[itex]^{2}[/itex] + s[itex]^{2}[/itex]))((e[itex]^{-st}[/itex]/a[itex]^{2}[/itex])(a sin at - s cos at))

= ((e[itex]^{-st}[/itex]/ (s[itex]^{2}[/itex] + a[itex]^{2}[/itex]))(a sin at - s cos at)

There is no way you could get the answer that you claim the book obtains: the Laplace transform of a function f(t) will not have a "t" in it, since t has be "integrated out". I hope you realize that you need to take a limit!
 
The integration limits on the LT integration are from 0 to ∞. Try these limits and see what you get. It's a definite integral, not an indefinite integral.
 
Applying limits on the last step was not a problem. I was getting wrong answer that is there was a mistake in sign at one place where Integration by parts is done second time. After fixing it I got the right answer which I have modified in post 1. After applying the limits I got the right answer.

e[itex]^{-s\infty}[/itex] becomes 0

and

e[itex]^{-s * 0}[/itex] becomes 1

sin 0 = 0

cos 0 = 1

First term of the equation after applying infinity becomes 0. Remaining is applying 0 as limit but a - sign appears before the equation.

- [ e[itex]^{-s * 0}[/itex] / (s[itex]^{2}[/itex] + a[itex]^{2}[/itex])(a sin a (0) - s cos a (0))

= - (1 / (s[itex]^{2}[/itex] + a[itex]^{2}[/itex])) ( - s)

= (s / (s[itex]^{2}[/itex] + a[itex]^{2}[/itex]))

Am I right?
 
Last edited:

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