1550078481-Ordinary_Differential_Equations__Roberts_

146 Ordinary Differential Equations

is .2 and with the parachute is 1.35. Compute the velocity of the

parachutist 20 seconds after jumping from the airplane. (HINT: You

need to solve two initial value problems. The solution of the first prob-

lem is the velocity of the parachutist for the first 10 seconds and the

solution of the second problem is the velocity of the parachutist for the

second 10 seconds. The final velocity of the first problem is the initial

velocity (initial condition) for the second problem.)

3.6 Mixture Problems

Suppose at time t = 0 a quantity qo of a substance is present in a container.
Also assume at time t = 0 a fluid containing a concentration C;,n(t) of the
substance is allowed to enter the container at the rate rin(t) and that the
mixture in the container is kept at a uniform concentration throughout by a
mixing device. Furthermore, assume for t 2'. 0 the mixture in the container
with concentration Cout(t) is all owed to escape at the rate rout(t). The problem
is to determine the amount, q(t), of substance in the container at any time.
See Figure 3.14.

ru.(t)

q(t)

roa (t)

Figure 3.14 Diagram for a One Tank Mixture Problem

Since the rate of change of the amount of substance in the container, dq / dt,
equals the rate at which the fluid enters the container times the concentration
of the substance in the entering fluid minus the rate at which the fluid leaves
the container t imes the concentration of the substance in the container, q(t)
must satisfy the initial value problem

(1)

dq

dt = rin(t)cin(t) - rout(t)Cout(t); q(O) = qo.