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Math 1320 Lab4 Instructor: ridget Fan Name: Due: 02,/11/16 in class Show all work including necessary graphs to get full credits! 1. Use the method of separation of variables to solve the following problems: (a) (y + xy)y’ Solve for y, where y’ = = 1 clx j(t+?c) i I Jç : (1’tfc (b) Find an equation of the curve that passes through (0, 1) and whose slope at (x, y) is 6x(y — 1)2/3. 6?c( -‘) f(j :I6,J?c (-) 3 (zj.C) j: (01 ) :) 1 3 -k-i C co Page 2 of 4 Lab4 Math 1320 2. A glass of hot water is cooling down with surrounding temperature of 72 degrees. The rate of change of the water temperature T(I) is proportional to the difference between T(t) and the surrounding temperature. Suppose at I = 0 the water temperature is 100 degrees and drops to 82 degrees after 10 minutes. (a) Set up the differential equation describing T(t). T’ft) k (T-2) (b) Solve the differential equation from part (a). (Hint: Your answer of T(t) should have two unknowns: the constant of proportionality and the arbitray constant c from integration.) (hlT.-i21 (Iq+c) :e (c) Use the two (b). flz. conditions given TeC+12 (j o) = c) by the problem to solve the two unknowns from part =) ecz2 too e° +11 2 (o ) CIn’I Iok ) ::) e 0 j 1oc (Qth& 1 IOW)I(f. Math 1320 Lab4 Page 3 of 4 3. A population P(t) has constant relative birth and death rates a and , respectively, and The a constant emigration rate m (a, and m are positive constants). Assume a > rate of change of the population at time t is modeled by . = kP — m, where k = a — (a) Find the solution of this equation that satisfies the initial condition P(0) —‘-—dy- cAk = Po. HCc&4v) ce (b) What condition on rn relative to kP 0 will lead to exponential expansion of the population? What condition on m relative to kP 0 will result in constant population? Population decline? Oj14tQ+iOr) Coflsfonl X ()Oii OplkAtn (c) In 1847, the population of Ireland was 8 million and the difference between the relative birth and death rates was 1.6o. As a result of the potato famine in the 1840s and 1850s, 210, 000 inhabitants per year emigrated from Ireland. Was the population expanding or declining? Cr r ,OVOoOO 000 okx4cw’ 9ec” e.. Page 4 of 4 Lab4 Math 1320 whether the sequence is 4. Determine (a) a = convergent or divergent and explain why. e {ic4 3e(ce Li fr,ø e e° :I 11 (b) a = Letb fri JJ— Je row ( c) a, a,, ia,, &j oqe i-i sin(nii) = a 3c (d) a, —b, hc ) fl Y°flt = ‘(n)O, ln(n) — ln(n — cr:- 1) — in(e. 1 COIAU4is ((X) i ..-.) f. (e) a,, = — i ( J2J’ /Z 1 - - V’-? - LO cWtTh: — — I (1j’l1.1 ‘ “ L’