Suppose that a certain population satisfies the initial value problem dy/dt = r

Question

Suppose that a certain population satisfies the initial value problem dy/dt = r (t )y ? k , y(0 ) = y0, where the growth rate r (t ) is given by r (t ) = (1 + sin t )/5, and k represents the rate of predation. (a) Supposethat k = 1/5. Plot y versus t for several values of y0 between1/2 and1. (b) Estimate the critical initial population yc below which the population will become extinct. (c) Choose other values of k and find the corresponding yc for each one. (d) Use the data you have found in parts (b) and(c) to plot yc versus k . Page(s): 62, Elementary Differential Equations and Boundary Value Problems, Ninth Edition by William E. Boyce, Wiley, John & Sons, Incorporated I really don't know what to do here for any of the questions

Explanation / Answer

dydt=r(t)y-k,y(0)=y0 where the growth rate r(t) is given by r(t)=1+sint5 and k represents the rate of predation. (a) Suppose that k=15 . Plot y versus t for several values of y0 between 1/2 and 1. (b) Estimate the critical initial population yc below which the population will become extinct. (c) Choose other values of k and find the corresponding yc for each one. (d) Use that data you have found in parts (a) and (b) to plot yc versus k ." I'm trying to apply the Method of Integrating Factors, but I'm stuck. Here's what I have: dydt-r(t)y=-k µ=exp[-15?(1+sint)dt]=exp(cost5-t5) y(t)=exp(t5-cost5)?-kexp(cost5-t5)dt y(t)=-kexp(t5-cost5)?exp(cost5-t5)dt???????????????????????????

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