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
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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.
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Example: Solve the second-order differential equation
d2 x
= −9x
dt2
where x(0) = 0 and x0 (0) = 12.
Example: Transform the second-order differential equation
d2 x
dx
−2
= 3x
2
dt
dt
into a system of first-order differential equations.
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