# Simultaneous Laplace transforms

by hurcw
Tags: laplace, simultaneous, transforms
 P: 20 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.
 P: 38 $$(s-4)X+2Y=2,5X-(s-2)Y=2.$$ $$X=\frac{2s}{s^2-6s+18},Y=-\frac{2s-18}{s^2-6s+18}.$$
 P: 20 Can you ellaborate a little please. Where did this all come from?
P: 38

## Simultaneous Laplace transforms

##\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,##$$X=\frac{2s}{s^2-6s+18}.$$
##5\times(1)-(s-4)\times(2):(10+(s-2)(s-4))Y=10-2(s-4),##$$Y=\frac{18-2s}{s^2-6s+18}.$$
 P: 20 Thats great, thanks alot. 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).
 P: 20 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
 P: 20 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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