# A Laplace Inverse Problem

• expi123
In summary, the conversation discusses trying to find the inverse of a given equation, but facing difficulty in factorizing the denominator. Different methods are suggested, such as using Cardano's method or performing partial fraction expansion. The conversation also touches on finding the type of differential equation and approximating it, but it is determined that there is not enough information to do so.
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?

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:
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.

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

Regards,

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

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?

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.

Is there a way to approximate the differential equation?

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?

## 1. What is a Laplace inverse problem?

A Laplace inverse problem is a mathematical problem that involves finding the original function from its Laplace transform. In other words, given the Laplace transform of a function, the goal is to determine the function itself.

## 2. Why is the Laplace inverse problem important?

The Laplace inverse problem is important in many fields of science and engineering, such as physics, electrical engineering, and control theory. It allows us to solve differential equations and analyze systems with complex behavior, making it a valuable tool in understanding and predicting real-world phenomena.

## 3. What are some applications of the Laplace inverse problem?

The Laplace inverse problem has numerous applications, including image and signal processing, circuit analysis, and control systems design. It is also used in solving problems related to heat transfer, fluid mechanics, and quantum mechanics.

## 4. What are the methods used to solve a Laplace inverse problem?

There are several methods for solving a Laplace inverse problem, including numerical methods, analytic methods, and integral inversion techniques. The choice of method depends on the complexity of the problem and the desired level of accuracy.

## 5. Are there any challenges associated with solving a Laplace inverse problem?

Yes, there are several challenges in solving a Laplace inverse problem, including the non-uniqueness of solutions, numerical instability, and the need for prior knowledge or assumptions about the original function. Additionally, the solution may not exist for certain types of functions or systems.

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