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A population has a periodic growth rate r(t) = A[1 + sin(t/(2*pi))], but otherwise follows the logistic population model with carrying capacity K. There is no threshold and the initial population is Yo = Y knot = K/2.

a. Modify the basic logistic equation for this population.

b. Use a change of variable z(t) = y/K to find a initial value equation in z(t). Then Solve for z(t).

So far I have:

Using the logistic growth equation: dy/dt = r(1-y/k)y

Leading to:

integral of [dy/((1-y/k)y)] = rt +c

substituting for r:

integral of [dy/((1-y/k)y)] = t*A[1 + sin(t/(2*pi))] + c

I can't really follow how to use the change of variables well so part b is the most confusing part.