How to Solve Simultaneous Laplace Transforms?

[hurcw]

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.

 

[Some Pig]

$$(s-4)X+2Y=2,5X-(s-2)Y=2.$$
$$X=\frac{2s}{s^2-6s+18},Y=-\frac{2s-18}{s^2-6s+18}.$$

 

[hurcw]

Can you ellaborate a little please.
Where did this all come from?

 

[Some Pig]

$\begin{cases}(s-4)X+2Y=2&...(1)\\5X-(s-2)Y=2&...(2)\end{cases}$
$(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),$$$Y=\frac{18-2s}{s^2-6s+18}.$$

 

[hurcw]

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).

 

[hurcw]

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

 

[hurcw]

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