Chap3StudentSolutions

02 people who die is the fraction of the total

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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 3-42 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 zero-excitation response and the zero-state 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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