1550078481-Ordinary_Differential_Equations__Roberts_

Applications of the Initial Value Problem y' = f(x, y); y(c) = d 155

Exercise 3. The y-axis and the line x = W > 0 represent the banks of a

river. The river flows in the negative y-direction with speed Sr. At time t = 0

a man starts walking from the origin along the negative y-axis with speed

Sm. At the same time a boat whose speed in still water is Sb is launched from

the point (W, 0). The boat is steered so that it is always headed toward the

man. Draw an appropriate figure for this problem, determine the differential

equation satisfied by the path of the boat, compute and graph the path of

the boat, if W = .5 mi, Sr = 3 mi/hr, Sm = 1 mi/hr, and (a) Sb = 2 mi/hr,

- (b) Sb= 3 mi/hr, and (c) Sb = 4 mi/hr. In each case, decide if the boat lands

on the opposite bank. Tell where the boat lands, when it lands. (HINT: Use

a "moving rectangular coordinate system" with origin at the man.)

Exercise 4. A rabbit starts at (0, a) and runs along the y-axis in the positive

direction with a constant speed of Sr. A dog starts at (b, 0) and pursues the

rabbit with speed sd. The dog runs so that he is always pointed toward the

rabbit. See Figure 3.20.

y

(0, a+ Sr t)

j sr Line tangent to the path of the dog

"/

"

(0, a)

(b, 0) x

Figure 3.20 Path of a Dog Pursuing a Rabbit

At time t the rabbit will be at the point (0, a+ srt) and the dog will be at

(x, y). Since the line between these two points is tangent to the path of the

dog, the slope of the line is

dy y-a - Srt

dx x

Multiplying by x, we get

(3) xy' = y - a - Sr t.