. V's-re go back to the logistic model for population dynamics (without harvesting), but we now allow the growth rate and carrying capacity to vary in time: (313 P E = amp (1 W). In this case the equation is not autonomous, so we can't use phase line analysis. \"Fe will instead nd explicit analytical solutions. (a) Show that the substitution 2 = 1 / P transforms the equation into the linear equation (12 + k(t)z = $8) (1)) Using your result in (a), show that if k is constant but Mr varies, the general solution is 6'\" PU) = C + f gdt (c) Similarly, show that if it! is constant but It varies, the general solution is A! P t = . ( ) 1 + CMe-IWW ((1) Consider the Special case where M is constant but I: decreases in time as k = e't. Suppose that the initial population is less than Mr. 'What happens to the population in the long run? Does it make sense