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John Michael
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Homework Statement
Find the steady-state oscillation of the mass–spring system
modeled by the given ODE. Show the details of your
calculations.
Homework Equations
1. y'' + 6y' + 8y = 130 cos 3t
2. 4y’’ + 8y’ + 13y = 8 sin 1.5t
The Attempt at a Solution
1. cos(3t) at the end means the basic angular frequency is 3 radians per second. Hence the steady-state oscillation frequency is 3/2pi Hz = 0.477 Hz.
2. solve for homogeneous differential equation,
4y'' + 8y' + 13y = 0
propose y = e^(ct)
y' = ce^(ct)
y'' = c²e^(ct)
substitute into mass-spring motion equation,
4y'' + 8y' + 13y = 0
4c²e^(ct) + 8ce^(ct) + 13ce^(ct) = 0
e^(ct)(4c² + 8c + 13) = 0
of course for unique solution, it must be e^(ct) ≠ 0
4c² + 8c + 13 = 0
c = -1 ± (3i)/2homogeneous solution is
y(t) = e^(-t) (A sin (3t/2) + B cos (3t/2))
where A and B is an arbitrary constants which dependent to boundary conditions
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