114_pdfsam_math 54 differential equation solutions odd

114_pdfsam_math 54 differential equation solutions odd -...

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Chapter 3 A = 1 (6000)7 ln ± 5000(3000 1000) 1000(5000 3000) ² = ln 5 42000 . Thus the formula (15) on page 95 of the text becomes p ( t )= p 0 p 1 p 0 +( p 1 p 0 ) e Ap 1 t = (1000)(6000) (1000) + (6000 1000) e (ln 5 / 42000)6000 t = 6000 1+5 1 t/ 7 . (3.1) In the year 2010, t = 2010 1980 = 30, and the estimated population of splake is p (30) = 6000 1 30 / 7 5970 . Taking the limit in (3.1), as t →∞ , yields lim t →∞ p ( t ) = lim t →∞ 6000 1 t/ 7 = 6000 1 + lim t →∞ 5 1 t/ 7 = 6000 . Therefore, the predicted limiting population is 6000. 15. Counting time from the year 1970, we have the following data: t 0 =0 ,p 0 = p ( t 0 ) = 300; t a = 1975 1970 = 5 a = p ( t a ) = 1200; t b = 1980 1970 = 10 b = p ( t b ) = 1500 . Since t b =2 t a , we use the formulas in Problem 12 to Fnd parameters in the logistic model. p 1 = ± (1200)(1500) 2(300)(1500) + (300)(1200)
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