Page 55 - Advanced engineering mathematics

1.5 Additional Applications 35

2

1

y
–3 –2 –1 0 1 2 3

–1

–2

x

FIGURE 1.10 Families of orthogonal trajectories in
Example 1.18.

This gives us
y

y = 2 = f (x, y)
x
as the differential equation of F. This means that F is the family of integral curves of y =2y/x.

The differential equation of the family of orthogonal trajectories is therefore
1 x

y =− =− .
f (x, y) 2y
This is a separable equation that can be written
2ydy =−xdx
with the general solution
1
2
2
y + x = c.
2
These curves are ellipses, and they make up the family G of orthogonal trajectories of F.
Figure 1.10 shows some of the ellipses in G and the parabolas in F.
A Pursuit Problem

In a pursuit problem, the object is to determine a trajectory so that one object intercepts
another. Examples are missiles fired at airplanes and a rendezvous of a shuttle with a space
station.
We will solve the following pursuit problem. Suppose a person jumps into a canal and
swims toward a fixed point directly opposite the point of entry. The person’s constant swim-
ming speed is v, and the water is moving at a constant speed of s. As the person swims, he
or she always orients to face toward the target point. We want to determine the swimmer’s
trajectory.
Suppose the canal has a width of w. Figure 1.11 has the point of entry at (w,0), and the
target point is at the origin. At time t, the swimmer is at (x(t), y(t)).

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October 14, 2010 14:9 THM/NEIL Page-35 27410_01_ch01_p01-42