Well, this topic is not in your textbook, but it's in the complimentary book and, besides, you are supposed to know whatever was covered in the lectures.
Assume that we have a one-parameter family of curves, i.e., a family of curves given by an equation F(x, y, a) = 0 with one parameter a. An orthogonal trajectory of the family is a curve which at each point is orthogonal to the curve of the family passing through this point. Typically, orthogonal trajectories also form a one-parameter family.
Orthogonal trajectories are found in two steps (see, e.g., Problem 3 in 2000 Midterm I).
First, the original representation F(x, y, a) = 0 should be converted to a differential equation (free of parameters!) whose integral curves are the curves of the given family. To do so, differentiate F(x, y, a) = 0 to get dF/dx+ y'dF/dy = 0 and eliminate  a from the system
Now in the equation G(x, y, y') = 0 replace y
with Y and y', with -1/Y', and solve the resulting
equation G(x, Y, -1/Y') = 0. Its integral
curves are the orthogonal trajectories of the original family.
The Wronskian W(x) of an n-tuple y1, ..., yn of solutions of an n-th order homogeneous linear differential equation