Correcting a Laplace Transform Problem

[jayanthd]

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$^{-st}$/(s$^{2}$ + a$^{2}$))(a sin at - s cos at)

Homework Equations

∫e$^{-st}$cos at dt

The Attempt at a Solution

u = e$^{-st}$

du = -se$^{-st}$ dt

Let dv = cos at dt

v = (sin at)/ a

Integrating by parts

∫e$^{-st}$cos at dt =

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

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

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

Rearranging

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

(e$^{-st}$ sin at / a) - (s/a$^{2}$)(e$^{-st}$ cos at)

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

∫e$^{-st}$cos at dt =

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

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

 

[Ray Vickson]

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$^{-st}$/(s$^{2}$ + a$^{2}$))(a sin at - s cos at)

Homework Equations

∫e$^{-st}$cos at dt

The Attempt at a Solution

u = e$^{-st}$

du = -se$^{-st}$ dt

Let dv = cos at dt

v = (sin at)/ a

Integrating by parts

∫e$^{-st}$cos at dt =

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

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

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

Rearranging

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

(e$^{-st}$ sin at / a) - (s/a$^{2}$)(e$^{-st}$ cos at)

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

∫e$^{-st}$cos at dt =

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

= ((e$^{-st}$/ (s$^{2}$ + a$^{2}$))(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!

 

[Chestermiller]

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.

 

[jayanthd]

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$^{-s\infty}$ becomes 0

and

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

= - (1 / (s$^{2}$ + a$^{2}$)) ( - s)

= (s / (s$^{2}$ + a$^{2}$))

Am I right?