More Orthogonal Trajectories Use Problem to determine the family of trajectories orthogona...

More Orthogonal Trajectories Use Problem to determine the family of trajectories orthogonal to each given family in Problem. (See Fig. 8.)

Problem

Orthogonal Trajectories

When a one-parameter family of curves satisfies a first-order DE, we can find another such family as solution curves of a related DE with the property that a curve from one family intersects each curve of the other family orthogonally (that is, at right angles: their respective tangent lines are perpendicular). Each family constitutes the set of orthogonal trajectories for the other.

See some intriguing pairs of orthogonaltrajectories.

For the following questions we use the customary independent variable x instead of t.

(a) Use implicit differentiation to show that the one- parameter family f(x, y) = c satisfies the differential equation dy/dx = −fx/fy, where fx = df/dx and fy = df/dy.

(b) Explain why the curves satisfying dy/dx = fy/fx are the orthogonal trajectories to the family in part (a).

(c) In Example 4, we found that the family x2 + y2 = c2 of circles with centers at the origin were the solution curves of the separable DE dy/dx = −x/y. Use this and part (b) to show that the family of orthogonal trajectories are the straight fines y = kx. (See Fig. 7. These families represent the electric field and equipotential lines around a point charge at the origin.)

Figure 7 Orthogonal families y = kx and x2 + y2 = c for Problem 48(c).

Figure 8 Orthogonal families y = c/x2 (in black) and ? (in color) for Problem

y = c/x2