How Do You Solve This Complex Laplace Inverse Problem?

[expi123]

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?

 

[Swapnil]

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.

 

[Swapnil]

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.

 

[Nenad]

Factor out the real root, then preform partial fractions on the denominator, split up your expression into two, and solve.

Regards,

Nenad

 

[expi123]

Thanks, I just could not find a standard method to factorise the denominator.

 

[robbin53]

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?

 

[CPL.Luke]

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.

 

[robbin53]

Is there a way to approximate the differential equation?

 

[CPL.Luke]

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?