What is wrong with my Laplace Transform inversion for y''+13y'-4y = 3exp(-t)?

In summary, the conversation is about finding the inverse of Y = s2 + 15s + 17 / [(s+1)(s2 + 13s - 4)] using PF expansion. The speaker has been getting wrong answers and their final equation, which is exp(-13/2t)[19/16cosh√(185)/2t + 13√(185)/74sinh√(185)/2t] - 3/16exp(-t) = y, does not agree with Runge-Kutta and Taylor approximation. They suspect that there is something wrong with the Bs + C term, since the numerator is a quadratic. The original differential equation is y"+13y'-
  • #1
maistral
240
17
I'm inverting this:

Y = s2 + 15s + 17 / [(s+1)(s2 + 13s - 4)]

I'm using PF expansion,

A/(s+1) + Bs + C/(s2 + 13s - 4), I however keep on getting wrong answers, seeing how Runge-Kutta and Taylor approximation disagrees with my final equation.

My final equation is:

exp(-13/2t)[19/16cosh√(185)/2t + 13√(185)/74sinh√(185)/2t] - 3/16exp(-t) = y, and it's wrong (considering Runge-Kutta and Taylor approximation disagrees with it).

Obviously, something's wrong. What did I miss? I'm starting to think that the second term is.. well, there's something wrong with it (Bs + C term). I mean, the numerator is a quadratic, therefore it can't be that simple...NOTE: This is NOT homework. I did this to merely tickle my head. The original differential equation is y"+13y'-4y = 3exp(-t), y(0) = y'(0) = 1.
 
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  • #2
And apparently according to this site, my inverse is:
http://wims.unice.fr/wims/wims.gif?cmd=getins&session=7D0AB1E6C0.3&special_parm=insert-1.gif&modif=1347637563

Crap, what is that i doing there?
 
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1. What is the Laplace Transform?

The Laplace Transform is a mathematical operation that transforms a function from the time domain to the complex frequency domain. It is commonly used in engineering and physics to solve differential equations.

2. What is Laplace Transform inversion?

Laplace Transform inversion is the process of transforming a function from the frequency domain back to the time domain. It is the inverse operation of the Laplace Transform and is used to solve differential equations and analyze systems in the time domain.

3. How is Laplace Transform inversion performed?

Laplace Transform inversion is performed by using the inverse Laplace Transform function, which is a complex integral. This integral can be solved using various methods such as partial fraction decomposition, contour integration, or the method of residues.

4. What are the applications of Laplace Transform inversion?

Laplace Transform inversion has various applications in engineering and physics. It is commonly used to solve differential equations and analyze systems in the time domain. It is also used in signal processing, control systems, and circuit analysis.

5. What are the limitations of Laplace Transform inversion?

One limitation of Laplace Transform inversion is that it can only be applied to functions that have a Laplace Transform. It also requires advanced mathematical techniques to solve the inverse integral, which can be challenging for some functions. Additionally, the inverse Laplace Transform may not exist for certain functions with complex poles or branch points.

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