1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Linear Equations with Constant Coefficients 301

(34a) x" = -HRy' +ER

(34b) y" = HRx'

where H is the intensity of the magnetic field and E is the intensity of an

electric field acting on the electron.

Exercise 4. Find the position of an electron as a function of time t, if

HR = 2, if ER = 3, and if the electron is initially at rest at the origin.

That is, solve the system (34) subject to the initial conditions: x(O) = 0,

x'(O) = 0, y(O) = 0, and y'(O) = 0. (HINT: Differentiate (34b) and substi-

tute (34a) into the resulting equation. Then solve the third order differential

equation in y , etc.)

Compartmental Analysis Many complex biological and physical pro-

cesses can be subdivided into several distinct phases. The complex process

can then be studied by analyzing each phase individually and the interaction

between the phases. Each phase or stage in the overall process is called a

compartment. It is assumed that material which moves from one compart-

ment to another does so in a negligible amount of time and that the material

itself is immediately dispersed throughout the entire compartment. A closed

compartmental system is one in which there is no input to or output from

any compartment in the system. An open system is one in which there is an

input to or output from at least one compartment in the system. Engineers

sometimes refer to compartmental systems as block diagrams.

Let Y 1 , Y2, ... , Yn denote the n separate compartments in a compart-

mental system. Each compartment is assumed to have a constant volume Vi

which may vary in size from compartment to compartment. At time t the

concentration of a particular substance Sin the compartment Yi is Yi(t). It is

assumed that at all times the substance is uniformly distributed throughout

each of the compartments. The rate of change of concentration of the sub-

stance in compartment i at time t, y~(t), is equal to the sum over all inputs

to the i-th compartment of the concentration of each input times the rate

of flow per volume of the input minus the concentration of the substance in

compartment i, Yi ( t), times the sum of the rates of flow per volume of output

from the compartment.

As an example, let Y1 ( t), Y2 ( t), and y3 ( t) denote the concentration of a

substance Sin compartments Y 1 , Y2, and Y3 at time t. And let a, b, c, d, e,

and f be the rates per volume at which the fluids containing the substance

S flow into and out of the compartments of the open compartmental system
shown in Figure 6.10. Also let u denote the constant concentration of the sub-
stance S in the fluid flowing into compartment Y 1. Under the assumptions
that the volume of each compartment remains constant, the time for material