Results 1 to 3 of 3

Math Help - Inverse Laplace Transforms

  1. #1
    Junior Member
    Joined
    Oct 2008
    Posts
    43

    Inverse Laplace Transforms

    Hey

    Having trouble with a couple of inverse Laplace transforms:

    L^{-1} {\frac{s}{(s-2)^2 + 9}}

    I know this uses cos{kt} and e^{at}, but can't figure out how to get that -2 out of there...

    Also,

    L^{-1} {\frac{s+4}{s^2 + 4s + 8}}

    I'm just lost on this one.

    Please help!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    From the tables...

    \mathcal {L}^{-1} \{\frac{1}{(s-b)^{2} + a^{2}}\} = \frac{e^{b t} \cdot \sin a t}{a} (1)

    ... so that is...

    \mathcal {L}^{-1} \{\frac{1}{(s-2)^{2} + 9}\} = \frac{e^{2 t} \cdot \sin 3 t}{3} (2)

    A basic properties of the LT is...

    \mathcal {L} \{f^{'} (t)\}= s\cdot \mathcal {L} \{f(t)\} - f(0) (3)

    ... so that...

    \mathcal {L}^{-1} \{\frac{s}{(s-2)^{2} + 9}\} = e^{2 t}\cdot (\frac{2}{3}\cdot \sin 3 t + \cos 3t) (4)

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    Is...

    \frac{s+4}{s^{2} + 4 s + 8} = \frac{s+4}{(s +2)^{2} + 4} (1)

    ... and because [see previous post...] is...

    \mathcal {L}^{-1} \{\frac{1}{(s+2)^{2} + 4}\} = \frac{e^{-2t}\cdot \sin 2t}{2} (2)

    ... and ...

    \mathcal {L}^{-1} \{\frac{s}{(s+2)^{2} + 4}\} = e^{-2t}\cdot (\cos 2t - \sin 2t) (3)

    ... we conclude that...

    \mathcal {L}^{-1} \{\frac{s+4}{(s+2)^{2} + 4}\} = e^{-2t}\cdot (\sin 2t + \cos 2t) (4)

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Laplace transforms? Inverse?
    Posted in the Advanced Applied Math Forum
    Replies: 4
    Last Post: March 13th 2012, 09:04 PM
  2. How to find these inverse Laplace transforms?
    Posted in the Advanced Applied Math Forum
    Replies: 3
    Last Post: March 29th 2011, 04:30 AM
  3. Inverse Laplace Transforms
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: October 13th 2010, 12:26 AM
  4. Inverse Laplace Transforms
    Posted in the Calculus Forum
    Replies: 2
    Last Post: August 11th 2009, 02:28 PM
  5. [SOLVED] inverse Laplace transforms
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: June 2nd 2009, 03:43 PM

Search Tags


/mathhelpforum @mathhelpforum