Answer the following questions. Please follow the format and write legibly to avoid confusion. (That is only 3 problems). (The last 3 pictures are for the answer for last number of letter a, b and c.) Thank you

1.

Find the general solution of the given differential equation, and use it to determine how solutions behave as t - co. = 7 cos(2t), t > 0 NOTE: Use c for the constant of integration. y Solutions converge to the function y =Suppose that a certain population has a growth rate that varies with time and that this population satisfies the differential equation dy dt = (0.3 + sint) NOTE: Round your answers to two decimal places. a) If y(0) = 1, find (or estimate) the time 7 at which the population has doubled. Choose other initial conditions and determine whether the doubling time 7 depends on the initial population. The doubling time is T = The doubling time Choose one depend on the initial population. b) Suppose that the growth rate is replaced by its average value 10 Determine the doubling time 7 in this case.l b) Suppose that the growth rate is replaced by its average value E' Determine the doubling time T in this case. c) Suppose that the term sint in the differential equation is replaced by sin(27rt); that is, the variation in the growth rate has a substantially higher frequency. What effect does this have on the doubling time T ? Some diseases (such as typhoid fever) are spread largel}r by carriers, individuals who can transmit the disease but 1vvho exhibit no overt symptoms. Let s and 1; denote the proportions of susceptibles and carriers, respectively, in the population. Suppose that carriers are identied and removed from the population at a rate [3, so dy _ = _ . 1 dt (33; ( ) Suppose also that the disease spreads at a rate proportional to the product of m and y; thus d3: E a) Determine y at any time t by solving equation (1) subject to the = cra:y. (2) initial condition y(0) = ya. at) = UH: a) Determine y at any time t by solving equation (1) subject to the initial condition y(0) = ya. W) = b) Use the result of part (a) to nd or at any time t by solving equation (2) subject to the initial condition 39(0) = mg. - Choose onevr c) Find the proportion of the population that escapes the epidemic by nding the limiting value of a: as t > ea. As If > co, the proportion of the population that escapes the epidemic is given by LIII'J a) Determine y at any time t by solving equation (1) subject to the initial condition y(0) = ya. b) Use the (a) to nd a: at any time t by solving equation (2) subj eityu) : condition mm) = mg. Choose one :1:(t) = \"Ho 1110530 the population that escapes the epidemic by (3) Find tll ' us of a: as t > oo. y (t ) Choose one V b) Use the Choose one to find x at any time t by solving equation (2) subj Coe @yo(Ite #t)/ B ndition *(0) = 20. x ( t) = To eryo(1te-#) / B c Find th Coe ayo(1-est) /B population that escapes the epidemic by finding fx as t - co. Coe ayo( 1-e-8t) / 8 Ast - d f the population that escapes the epidemic Co e yo(1-e-#)/8 one Co e yo(1-eat)/Ba) Determine y at any 1) subject to the initial condition y( y (t ) = Choose one Choose one b) Use the result of pa Co e"yo/ B by solving equation (2) subject to the in x(t) = Choose one Coe ayo/B c) Find the proportion pes the epidemic by finding the limiting 0, i.e. no individual As t - co, the propo escapes the pandemic. escapes the epidemic is given by the limiting value of x for t - oo does not exist