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A Laplace Inverse Problem

  1. Nov 12, 2006 #1
    I am trying to do the inverse of the foll. eqn. to no avail:

    s+1/(s^3 + s + 1)

    I cannot factorise the denominator.
    My only alternative was to use the defintion of Laplace and try to integrate the equation. Still I could not. Can anyone provide some hints?
  2. jcsd
  3. Nov 12, 2006 #2
    That function in the denominator is cubic so it atleast has one real root. After graphing it, I can see that it has 1 real root. Hence, it has two complex roots. My guess would be to write that polynomial as a product of one real factor and two complex factors. I have no idea how to do that though.

    It is a tough one though! I hope someone else in this forum is able to help you.
    Last edited: Nov 12, 2006
  4. Nov 13, 2006 #3
    Actually, you can use the Cardano's method to find the exact roots.

    Numerically, though, the roots are:
    .341 + 1.16j; .341-1.16j; and -.682

    Now you can treat these complex numbers just like any other real numbers and do the partial fraction expansion.
  5. Nov 19, 2006 #4
    Factor out the real root, then preform partial fractions on the denominator, split up your expression into two, and solve.


  6. Nov 22, 2006 #5
    Thanks, I just could not find a standard method to factorise the denominator.
  7. Dec 7, 2006 #6
    Hi, I need some help!

    I performed inverse laplace on (s^2+5s+6)/(s^3-s^2+2) and got 3/5*e^t*cos(t)+29/5*e^t*sin(t)+2/5*e^-t. My question is how do you take it back to the differential equation and how do you tell what type of differential equation it is?
  8. Dec 7, 2006 #7
    hmm why would you want to?

    if you used a laplace transform than it must have been a linear differential equation, combined with an initial value problem.

    also because the highest power of s is s^3 I'd say that it couldn't have been more than third order. (although there might be special circumstances that I'm missing.)

    I don't believe there is enough information to easily reconstruct the differential equation.
  9. Dec 7, 2006 #8
    Is there a way to approximate the differential equation?
  10. Dec 7, 2006 #9
    none that I know of, part of your problem is that you don't know what the original driving function was, or the initial conditions.

    because a laplace transform depends on both of these things its impossible to determine what the differential equation was.

    but once again why would you want to reconstruct the differential equation?
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