# Inverse Laplace

Discussion in 'Homework Help' started by Hasnatsaeed, Dec 11, 2010.

1. ### Hasnatsaeed Thread Starter New Member

Sep 23, 2010
27
0
Hi..i need to determine the inverse laplace of F(s)=((s^2)-(∏^2))/((s^2)+(∏^2))^2..........by either differentiation or integration of Transforms method..no other methods..i know thee inverse laplace of this s-domain function is t cos(∏t) but i cant figure out a way to show this through either of the two methods mentioned.plz help thanks

2. ### tyblu Member

Nov 29, 2010
199
16
This belongs in the math forum.

$
F(s) = \left(\frac{s^2-\pi^2}{s^2+\pi^2}\right)^2 \\
\\
\text{Let } Y(s) = \frac{s^2-\pi^2}{s^2+\pi^2}
= \frac{s^2}{s^2+\pi^2} - \frac{pi^2}{s^2+\pi^2} \\
\\
\mathscr{L}^{-1}\{Y(s)\}
= \mathscr{L}^{-1}\{\frac{s^2}{s^2+\pi^2} - \frac{pi^2}{s^2+\pi^2}\}
= \mathscr{L}^{-1}\{\frac{s^2}{s^2+\pi^2}\} - \mathscr{L}^{-1}\{\frac{pi^2}{s^2+\pi^2}\}
$

You do the next part, and indicate where you get stuck.

3. ### Hasnatsaeed Thread Starter New Member

Sep 23, 2010
27
0
its F(s)=((s^2)-(∏^2)) / ((s^2)+(∏^2))^2....the denominator is squared not the whole expression and i am supposed to determine the inverse laplace through either differentiation or integration of laplace.i.e by either using......L{tf(t)}= -F'(s)...or......L{f(t)/t}=∫F(s).ds(the limit is from 0 to ∞)

4. ### Hasnatsaeed Thread Starter New Member

Sep 23, 2010
27
0
i tried to put it into the math section but due to double threads it was closed..so i cant actually

Apr 5, 2008
15,536
2,302
Hello,

I said, you can continue here.

Bertus

6. ### tyblu Member

Nov 29, 2010
199
16
Squaring just the denominator, then:

$
F(s) = \frac{s^2-\pi^2}{\left(s^2+\pi^2\right)^2} \\
= \frac{s^2}{\left(s^2+\pi^2\right)^2} -
\frac{\pi^2}{\left(s^2+\pi^2\right)^2} \\
\\
\mathscr{L}^{-1}\{F(s)\}
= \mathscr{L}^{-1}\{\frac{s^2}{\left(s^2+\pi^2\right)^2} -
\frac{pi^2}{\left(s^2+\pi^2\right)^2}\}
= \mathscr{L}^{-1}\{\frac{s^2}{\left(s^2+\pi^2\right)^2}\} -
\mathscr{L}^{-1}\{\frac{pi^2}{\left(s^2+\pi^2\right)^2}\}
$

You do the next part, and indicate where you get stuck. There are common identities available in your textbooks for the solutions, if you want to check. If you could reply in either LaTeX like I have or some other way that makes it easy to read, you would increase the chances of getting a reply.

7. ### Hasnatsaeed Thread Starter New Member

Sep 23, 2010
27
0
To use any one of the methods mentioned i have to either first differentiate it or integrate it with respect to s and then take their inverse laplace and then multiply or divide by t depending upon whether i am following differentiation method or integration..thats the approach i am supposed to follow. you are just telling me to get the answer out straight foreward by taking inverse laplace.

8. ### Hasnatsaeed Thread Starter New Member

Sep 23, 2010
27
0
if i differentiate i get this.now the inverse laplace of this is equal to -tf(t)...now how can i show that f(t)=tcos(pi.t)?

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