- #1
RJLiberator
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Homework Statement
Verify that Ae^(-βt)cos(ωt) is a possible solution to the equation:
d^2(x)/dt^2+ϒdx/dt+(ω_0)^2*x = 0
and find β and ω in terms of ϒ and ω_0.
Homework Equations
N/a, trig identities I suppose.
The Attempt at a Solution
I think this is simply a 'plug and chug' type equation, but I'm having alll sorts of difficulty canceling things.
I first calculated the first and second derivative of the given possible solution.
First derivative = Ae^(-βt)*(-ωsin(ωt))+(-Aβe^(-βt)cos(ωt))
Second derivative = -Ae^(-βt)ω^2cos(ωt)+Aβe^(-βt)ωsin(ωt)+Aβ^2e^(-βt)cos(ωt)+Aβe^(-βt)ωsin(ωt)
I then plugged them into their respective spots into the equation and tried to simplify.
I factored and divided both sides by Ae^(-βt).
I am at:
2βωsin(ωt)+β^2cos(ωt)-ϒ(ωsin(ωt)+βcos(ωt))+(ω_0)^2cos(ωt)-ω^2cos(ωt) = 0
The problem I am having is visualizing how the terms can cancel.
How does a term with ϒ cancel with terms with ω and terms with (ω_0)^2
I have to be missing something.