1550078481-Ordinary_Differential_Equations__Roberts_

302 Ordinary Differential Equations

to flow from one compartment to another is negligible, and the fluid in each
compartment has uniform concentration, the concentrations Yi satisfy the
following system of differential equations

(35a)

(35b)

(35c) y~ = cyz - y3(d + e).

J L

Figure 6.10 An Open Compartmental System

Solving equation (35a) for y 2 and differentiating, we find

Substituting these expressions for y 2 and y~ into equation (35b) and multi-

plying by f , we get

y~ +by~ = bfy1 + efy3 - (c + f)(y~ - au+ by1).

Solving this equation for y3 and differentiating, we obtain

Y3 = [y~ + (b + c + f)y~ + bcy1 - (c + f)au]/(ef)

and

y~ = [Yi3) + (b + c + f)y~ + bcy~]/(ef).

Substituting from the above expressions for y2, y3, and y~ in equation (35c),
we find y 1 satisfies the single differential equation