1550078481-Ordinary_Differential_Equations__Roberts_

148

140

120

100

80

y(x)

60

40

2 0

0

Ordinary Differential Equations

0 1 2 3 4 5 6

x

Figure 3.15 Numerical Approximation to the IVP:

y' = 16 - 3y/(165 + 5x); y(O) = 20

I EXAMPLE 2 Two Tank Mixture Problem

7

A 150 -gallon tank is initially filled with a 603 dye solution. A 103 dye
solution flows into the tank at a rate of 5 gal/min. The mixture flows out
of the tank at the same rate into a 75 gallon tank which was initially filled
with pure water. The mixture in the second tank flows out at the rate of
5 gal/min. Compute and graph the amount of dye in the second tank on the
interval [O, 125 ]. When is the amount of dye in the second tank a maximum?
What is the maximum amount of dye in the second tank?

SOLUTION

Let q1 ( t) and q2 ( t) denote the number of gallons of dye in tanks 1 and 2 at
time t, respectively. A diagram for this example is displayed in Figure 3.16.

The initial number of gallons of dye in tank 1 at time t = 0 is q 1 (0) =

603 x 150 gal = 90 gal. Since the second tank is initially filled with pure

water, q2(0) = 0 gal.

For tank 1, we have run(t) = ri_out(t) = 5 gal/min, cun(t) = 103 = .1,

and Ci_ 0 ut(t) = q1 (t)/150. So the number of gallons of dye in tank 1 must

satisfy the differential equation