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A dynamical system and Impulse function

Hi, I have been asked this question...

A dynamical system is modelled by the equation

x¨(t)+3x˙(t)+2x(t)=f(t)\ddot{x}(t) + 3\dot{x}(t) + 2x(t) = f(t)

where f(t) is an impulse of strength 3 units applied at t=4 and is subject to the initial conditions

x(0)=2x(0) = 2 and x˙(0)=−4\dot{x}(0) = -4

Find an expression for x in terms of t

I am little confused as to how I would start. I know i need to write it f(t) as the dirac delta function, but I don't know how to do this. I can do the rest of the calculations.
(edited 13 years ago)
Reply 1
Avatar for sarah_vickers
sarah_vickers
OP
Original post by PerigeeApogee
Have you studied Laplace Transforms?

If so, that would, without doubt, be the easiest way to approach the problem. If not, let me know.


yeah thats what I'm going to use to solve this. But in lectures, we have always written f(t) in the form of the dirac delta function. I know how to solve it but I just need to change f(t) to dirac delta.
Reply 2
Avatar for DFranklin
DFranklin
18
Well, δ(t)\delta(t) is an impulse of strength 1 unit applied when t = 0...
Reply 3
Avatar for sarah_vickers
sarah_vickers
OP
oh i see.
Reply 4
Avatar for DFranklin
DFranklin
18
Original post by PerigeeApogee
Or, as Franklin says.... f(t)=4δ(t+4) f(t) = 4 \delta (t+4)
At least a couple of errors there, but note also that I wanted the OP to find the expression for herself.
(edited 13 years ago)
Reply 5
Avatar for sarah_vickers
sarah_vickers
OP
Original post by DFranklin
At least a couple of errors there, but note also that I wanted the OP to find the expression for herself.


would it not be,

f(t)=3δ(t−4)f(t) = 3 \delta (t-4)
(edited 13 years ago)
Reply 6
Avatar for DFranklin
DFranklin
18
Yes, exactly.
Reply 7
Avatar for sarah_vickers
sarah_vickers
OP
Original post by DFranklin
Yes, exactly.


Thanks very much. I can do the rest of the calculations from ghere :smile: xxx

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