How to Solve Simultaneous Laplace Transforms?

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Discussion Overview

The discussion revolves around solving a system of simultaneous differential equations using Laplace transforms. Participants are exploring the steps involved in deriving expressions for the variables involved, as well as the subsequent inverse Laplace transforms.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant presents the system of equations and initial conditions, seeking guidance on how to proceed with the Laplace transform.
  • Another participant provides transformed equations for the variables X and Y, suggesting specific expressions derived from the initial equations.
  • A request for clarification is made regarding the derivation of the expressions for X and Y, indicating a desire for deeper understanding.
  • Further mathematical manipulation is shown, leading to expressions for X and Y, with an emphasis on the steps taken to arrive at these results.
  • One participant expresses curiosity about the origins of specific terms in the expressions for X and Y, indicating a need for clarification on the calculations involved.
  • Another participant confirms understanding of the terms and raises a question about the nature of the roots in the expressions, suggesting they may be complex.
  • A participant seeks assistance in finding the inverse Laplace transforms of X and Y, proposing specific forms for these transforms and expressing gratitude for the forum's support.

Areas of Agreement / Disagreement

Participants generally agree on the mathematical manipulations performed but express uncertainty regarding the origins of specific terms and the nature of the roots involved. The discussion remains unresolved regarding the final forms of the inverse Laplace transforms.

Contextual Notes

Some participants express uncertainty about the derivation of terms in the equations, and there is a lack of consensus on the complexity of the roots involved in the expressions for X and Y. The discussion does not resolve the steps needed to find the inverse Laplace transforms definitively.

hurcw
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I have to try and solve the following simultaneous Laplace transform problem and don't really know which path to take can someone give me a nudge in the right direction please.

dx/dt=4x-2y & dy/dt=5x+2y given that x(0)=2, y(0)=-2
this is what i have so far for dx/dt=4x-2y
sx-x(0)=4x-2y
sx-2=4x-2y
(s-4)x+2y=2

And for dy/dt=5x+2y
sy-y(0)=5x+2y
sy+2=5x+2y
(s-2)y-5x=-2
Not really sure where to go from here, or even if this is correct.
 
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[tex](s-4)X+2Y=2,5X-(s-2)Y=2.[/tex]
[tex]X=\frac{2s}{s^2-6s+18},Y=-\frac{2s-18}{s^2-6s+18}.[/tex]
 
Can you ellaborate a little please.
Where did this all come from?
 
##\begin{cases}(s-4)X+2Y=2&...(1)\\5X-(s-2)Y=2&...(2)\end{cases}##
##(s-2)\times(1)+2\times(2):((s-4)(s-2)+10)X=2(s-2)+4,##[tex]X=\frac{2s}{s^2-6s+18}.[/tex]
##5\times(1)-(s-4)\times(2):(10+(s-2)(s-4))Y=10-2(s-4),##[tex]Y=\frac{18-2s}{s^2-6s+18}.[/tex]
 
Thats great, thanks a lot.
Just out of interest where has the 2s in X come from and the 18 - 2s in Y come from, i can work out the bottom lines. sorry if i appear stupid but it is 5.20am.
From there i can use partial fractions to determine the inverse Laplace transform (I think anyway).
 
I get the 2s & the 18-2s.
Am i correct in thinking these sre complex roots and by definition are quite complex to solve especially the 18-2s one.?
any help is appreciated
 
I need to then try and find the inverse Laplace transform of X & Y can anyone assist me in telling me if i am close with:-
X=2e^(-3t)*cosh3t
Y=e^(-18t)-2e^(-3t)*cosh3t

This forum has been more than helpful so far and is highy recommended
 

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