Sorry, HTML is a horrible language, so formulas and spacing are not going to come out right...

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

(The latter is to be regarded as an algebraic system of two equations in four unknowns

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 *y*1, ..., *yn*
of solutions of an *n*-th order homogeneous linear differential equation

satisfies the equation

Hence, it is given by

and in order to find