Applications of the Initial Value Problem y' = f(x, y); y(c) = d 153
the second term represents the amount of pollutants entering Ontario
from non-Erie sources, and the third term represents the amount of
pollutants leaving Ontario through outflow. Assuming the volume of
Lake Ontario, Va, is constant, we find, by dividing the previous equation
by Va, that the concentration of pollution in Lake Ontario, ca, satisfies
the differential equationdca reCe +re - raCa
dt VaCompute and graph the concentration of pollution in Lake Ontario ifca(O) = 0.3%, ce(t) = 0.4%e-t/lO and c(t) = .08%. How many years
will it take for the concentration of pollution in Lake Ontario to reach
0.2%? 0.1%?3. 7 Curves of Pursuit
Interesting problems result when one tries to determine the path that one
object must take in order to pursue, and perhaps capture, a second object
when either or both objects move according to specified constraints.
Exercise 1. A boy attaches a stiff rod of length L to a toy boat. The boy
places the boat at the edge of a rectangular pool at (L, 0) and then moves to
the corner of the pool at (0, 0). As the boy walks along the other edge of the
pool- the y-axis, the boat glides through the water after him. See Figure 3.18.
The path followed by the boat is called a tractrix (Latin tractum, drag).
yPath of Boat
p(0, 0) (L, 0) xFigure 3.18 The Tractrix