-Mathematical Modeling

#1) A patient is given a dosage Q of a drug at regular intervals of time T. The concentration of the drug in the blood has been shown experimentally to obey the law dC/dt = -keC

a. If the first dose is administered at t = 0 hr, show that after T hr have elapsed, the residual R1 = ln(kT + e-Q) remains in the blood.

b. Assuming an instantaneous rise in concentration whenever the drug is administered, show that after the second dose and T hr have elapsed again, the residual R2 = (-ln)(kT/1-eQ) remains in the blood.

c. Show that the limiting value R of the residual concentrations for doses of Q mg/ml repeated at intervals of T hr is given by the formula R = (-ln)(kT/1-eQ)

d. Assuming the drug is ineffective below a concentration L and harmful above some higher concentration H, show that the dose schedule T for a safe and effective concentration of the drug in the blood satisfies the formula T = (1/k)(e-L – e-H) where k is a positive constant.

#2) dy/dx = y^2 – 2y

a. Identify the equilibrium values. Which are stable and which are unstable?

b. Construct a phase line. Identify the signs of y’ and y”.

c. Sketch several solution curves.