You can see that the amount of medication enters the therapeutic window quickly, but goes below the minimum near t=2.5 and exceeds the maximum near t=3.5. During the first time period (0 ≤ t ≤ ½), the medication is being released into the system, while there is also decay. In the second time period (½ ≤ t ≤ 3), the medication is no longer being released in the system, and decay is taking place. In the third time period (3 ≤ t ≤ 3.5), a second dose of the time release medication is given, etc. If dy/dt=f(t,y), you can write f(t,y) as a piecewise defined function with four pieces for each of the four intervals in which something different is happening. Recall from your readings that piecewise defined functions can be written as a single function by incorporating step functions and window functions (Section 7.6). In this discussion, you’ll examine other potential release and administration times to see if they stay within the therapeutic window. Assume the following: The decay rate for the medication in the body is equal to the amount of medication present. That is, assume that the decay constant is 1. The therapeutic window is y between 0.1 and 0.8. Units in this discussion are grams (g) and hours (h). You will choose your data from the lists below for the constant release time b and administration time T (both in hours). The time release medication contains 1 g of medication per dose. Each dose will release for b h at a constant rate of 1/b g/h. For example, if b=½, then 1/b = 2, and the medication will be released in ½ h at a rate of 2 g/h. Two identical doses of medication will be administered, one at time t=0 and one at administration time T. Your post should contain the following information: The initial value problem for y(t). The function y(t) measures the amount of medication in the body at time t. (The setup is rate = rate in – rate out.) A detailed description of the solution of the initial value problem, including Laplace transform and inverse Laplace transform calculations. The graph of the solution over the interval 0 ≤ t ≤ 2T. An analysis of the relation of the solution y(t) to the therapeutic window. You will need to address the following in your analysis. Does y(t) enter the window in the first ½ h? If it doesn’t enter the window quickly enough or leaves the window at other times, make recommendations about how you might change b or T based on your solution and its graph. If it stays within the window for 0 ≤ t ≤ 2T, you could suggest changes to b and T that would allow y(t) to stay within the window over a longer time. Explain your reasoning. My b value is 1 and my T value is 3