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Unformatted text preview: (τ ) + e −6.667τ [13.333 cos(2.357τ ) − 16.497 sin(2.357τ )] u(τ ) u( t − τ ) dτ
−∞ ∞ { } z( t) = 0.15 ∫ −δ (τ ) + e −6.667τ [13.333 cos(2.357τ ) − 16.497 sin(2.357τ )] u( t − τ ) dτ
0− For t < 0, z( t) = 0.
For t > 0,
using
ax
∫ e sin(bx )dx = e ax
[a sin(bx ) − b cos(bx )]
a2 + b2 e ax
∫ e cos(bx )dx = a2 + b2 [a cos(bx ) + b sin(bx )]
ax we get t e −6.667τ
13.333 50 [−6.667 cos(2.357τ ) + 2.357 sin(2.357τ )] z( t) = −0.15 u( t) + 0.15 e −6.667τ −16.497 50 [−6.667 sin(2.357τ ) − 2.357 cos(2.357τ )] −
0 or e −6.667 t
13.333
[−6.667 cos(2.357t) + 2.357 sin(2.357t)] 50 e −6.667 t z( t) = −0.15 u( t) + 0.15 −16.497
[−6.667 sin(2.357t) − 2.357 cos(2.357t)] 50 −13.333 −6.667 + 16.497 −2.357 50
50 { } z( t) = −0.15 u( t) + 0.15 e −3.333 t [2.812 sin(2.357 t) − cos(2.357 t)] + 1 u( t)
or z( t) = 0.15e −3.333 t [2.812 sin(2.357 t) − cos(2.357 t)] u( t)
z(t)
0.1
2 0.2 Solutions 342 t M. J. Roberts  8/16/04 58. As derived in the text, a simple pendulum is approximately described for small angles, θ ,
by the differential equation,
mLθ ′′ ( t) + mgθ ( t) ≅ x( t)
where m is the mass of the pendulum, L is the length of the massless rigid rod supporting
the mass and θ is the angular deviation of the pendulum from vertical.
(a) Find the general form of the impulse response of this system.
After time, t = 0 he impulse is an undamped sine function whose (radian) frequency
g
is
.
L 59. Pharmacokinetics is the study of how drugs are absorbed into, distributed through,
metabolized by and excreted from the human body. Some drug processes can be
approximately modeled by a “one compartment” model of the body in which V is the
volume of the compartment, C( t) is the drug concentration in that compartment, ke is a
rate constant for excretion of the drug from the compartment and k0 is the infusion rate
at which the drug enters the compartment.
(a)
Write a differential equation in which the infusion rate is the excitation and the drug
concentration is the response.
mg
(where “ l ” is
(b)
Let the parameter values be ke = 0.4 hr −1, V = 20 l and k0 = 200
hr
mg
the symbol for “liter”). If the initial drug concentration is C(0) = 10
, plot the drug
l
concentration as a function of time (in hours) for the first 10 hours of infusion. Find the
solution as the sum of the zeroexcitation response and the zerostate response.
(a)
The differential equation equates the rate of increase of drug in the compartment to
the difference between the rate of infusion and the rate of excretion.
V d
(C(t)) = k0 − Vke C(t)
dt 60. At the beginning of the year 2000, the country, Freedonia, had a population, p, of 100
million people. The birth rate is 4% per annum and the death rate is 2% per annum,
compounded daily. That is, the births and deaths occur every day at a uniform fraction
of the current population and the next day the number of births and deaths changes
because the population changed the previous day. For example, every day the number of
0.02
people who die is the fractio...
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 Spring '09
 ATOUSA

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