Help with Inverse Laplace Problems!

[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 can't get the a and w right! :(

with 2.
I know that it needs to be expanded using partial fractions (A+Bs+C)
But I can't 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!

 

[Office_Shredder]

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

 

[vela]

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?

 

[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 :)

 

[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?

 

[vela]

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.

 

[cattmiow]

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

 

[vela]

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

 

[gomunkul51]

Thanks to everyone for your help.
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?

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) :)