System of Differential equations using Laplace.

rellsun

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May 2, 2012
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Hey everyone i have a system of Differential equations that must be solved using Laplace transformation thats giving me trouble.
Heres the problem :9.3.jpg. all i have so far is this: x'=5x+5y+2z, y'=-6x-6y-5z, z'=6x+6y+5z.
 
x'=5x+5y+2z, y'=-6x-6y-5z, z'=6x+6y+5z.
Its been many years, but the Laplace( x') = s * Laplace( x) if my memory serves me correctly. Thus, rewrite your equations like this: sX = 5X + 5Y + 2 Z, sY = -6X - 6Y - 5Z, and so on. Now collect terms on the left, e.g.
(s - 5)X = 5Y + 2Z ==> X = (5Y + 2Z) / (s -5). Do this same thing for Y & Z. Now you have a system of Laplace equations. Solve this system of 's' dependent variables and you'll have an answer.
 
CurveGuy is correct about the method but assumed the initial conditions are all 0. For example [MATH]\dot X = sX-x(0)[/MATH]. This leads to the linear system:

[MATH]\begin{pmatrix} sX-x(0)\\sY-y(0)\\sZ-z(0) \end{pmatrix} = \begin{pmatrix} 5 & 5 & 2 \\ -6 & -6 & -5 \\ 6 & 6 & 5 \end{pmatrix} = \begin{pmatrix} X \\ Y \\ Z \end{pmatrix} [/math]where the initial conditions might not be [MATH]0[/MATH]. You can solve the system by any favorite method you choose to get the transforms [MATH]X,~Y,~Z[/MATH] after which you need to find their inverses.
That [MATH]6[/MATH] with a line above it should be [MATH]-6[/MATH]. There appears to be a problem with Latex rendering.
 
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