Homework Help: LaPlace Probs!

1. Aug 5, 2010

cattmiow

Hi!
Im having a little (a lot) of trouble with some inverse Laplace problems.
If anyone can help, I would really appreciate it!!!

1. 10s/(s^2+256)^2

2. 6s+6/(s+14)(s^2+4)

With 1. I have tried countless ways of rearranging etc. getting (s^2+16^2)^2 to try to work it out that way...
Im thinking it should either fit with either
sin(wt)-atcos(wt)
OR
sin(wt)+at cos (wt)

but I cant get the a and w right! :(

with 2.
I know that it needs to be expanded using partial fractions (A+Bs+C)
But I cant work out what the numerator is meant to me
I think the deniminator is s+7, (s+7)^2, (s+7)^3, (s+7)^4
And that I can bring that out as a constant and have e^-7t before those terms, but once again, Help!!

Thanks again!!

2. Aug 5, 2010

Office_Shredder

Staff Emeritus
For number 1, try

$$\frac{10}{s^2+256} \frac{s}{s^2+256}$$

Think about what you can take the Laplace transform of to get multiplication of two functions

3. Aug 5, 2010

vela

Staff Emeritus
Another approach to the first one is to use the fact that

$$\frac{d}{ds}\left(\frac{1}{s^2+256}\right) = -\frac{2s}{(s^2+256)^2}$$

On the second problem, is it

$$6s + \frac{6}{(s+14)(s^2+4)}$$

or

$$\frac{6s+6}{(s+14)(s^2+4)}$$

and where are you getting (s+7) and its higher powers in the denominator from?

4. Aug 5, 2010

gomunkul51

1. Convolution. or use the template {2as/(s^2 + a^2)^2} -> t sin(at).

2. Partial Fraction Decomposition:

(6s + 6)/(s + 14)(s^2 + 4) = A/(s + 14) + (Bs + C)/(s^2 + 4)

*A,B,C = const.

Then use templates:

{1/(s - a)} -> exp(at)
{a/(s^2 + a^2)} -> sin(at)
{s/(s^2 + a^2)} -> cos(at)

Good luck :)

5. Aug 6, 2010

cattmiow

Thanks to everyone for your help.
I worked the first one out as
5/16*t*sin(16*t)

:)

Still struggling with the second one.
I have tried to input it into Matlab for help, but struggling with the Dirac component.
(6*s+6)/((s+14)*(s^2+4))

returns 12*dirac(1,t)-84*dirac(t)+1200*exp(-14*t)

Any ideas on what this means?

6. Aug 6, 2010

vela

Staff Emeritus
The result Matlab is giving you isn't correct, so don't waste your time trying to figure out what it means.

Show us your work for the partial fraction expansion of the transform.

7. Aug 6, 2010

cattmiow

I got A = -39/100
B = 39/100
and
C = 21/100
am i on the right track?

8. Aug 6, 2010

vela

Staff Emeritus
Your values for A and B are correct, but C should be 54/100.

9. Aug 6, 2010

gomunkul51

don't pay attention to the Dirac function, answer it as you know.
the same answer can be written in different ways.

again, {1/(s - a)} -> exp(at) :)