Section 11.2: Linear Systems (Applications) Compartmental models describe the flow between compartments or states of a system. In this section, we consider a general two-compartment model, where x1 (t) and x2 (t) denote the amounts of matter in compartments 1 and 2 at time t ≥ 0, respectively. The general two-compartment model is illustrated by the diagram below. I x1 ax1 bx2 x2 dx2 cx1 The following linear system of differential equations describes the flow into and out of each compartment. dx1 = I − (a + c)x1 + bx2 , dt dx2 = ax1 − (b + d)x2 . dt Example: Draw the compartmental diagram corresponding to the following system of linear differential equations dx1 = −0.4x1 + 3x2 dt dx2 = 0.2x1 − 3x2 . dt 1 Example: Find the system of differential equations corresponding to the compartmental diagram below and analyze the stability of the equilibrium (0, 0). x1 0.4 x1 1.2 x2 x2 0.3x1 Example: Find the system of differential equations corresponding to the compartmental diagram below and analyze the stability of the equilibrium (0, 0). x1 0.1x1 1.7 x1 0.6 x2 x2 0.3x2 2 Compartmental models are used in pharmacology to study how drug concentrations change within an individual’s body. Example: Suppose that a drug is administered to a person in a single dose, and assume that the drug does not accumulate in body tissue, but is excreted through urine. Denote the amount of drug in the body at time t by x1 (t) and in the urine at time t by x2 (t). If x1 (0) = 6 mg and x2 (0) = 0 mg, find x1 (t) and x2 (t) if dx1 = −0.3x1 (t). dt Example: Suppose that a drug is administered to a person in a single dose, and assume that the drug does not accumulate in body tissue, but is excreted through urine. Denote the amount of drug in the body at time t by x1 (t) and in the urine at time t by x2 (t). If x1 (0) = 6 mg and x2 (0) = 0, find a system of differential equations for x1 and x2 if it takes 20 minutes for the drug to be at one half of its initial amount in the body. 3 Second-Order Linear Differential Equations Example: Transform the second-order differential equation d2 x dx = 3x − 2 dt2 dt into a system of first-order differential equations. 4