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 of the other family orthogonally. Each family constitutes the set of orthogonal trajectories for the other.
For the following questions we use the customary independent variable x instead of r.
(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 = (f/(x and fy (f/(y
(b) Explain why the curves satisfying dy/dx = fy/fx are the orthogonal trajectories to the family in part (a)
(c) 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 lines y = kx. (See Fig. 1.3.7. These families represent the electric field and equi-potential lines around a point charge at the origin.)