Chapter 8.2, Problem 33P

Find the orthogonal trajectories of each of the following families of curves. In each case, sketch or computer plot several of the given curves and several of their orthogonal trajectories. Be careful to eliminate the constant from $y^{\prime }$ for the original curves; this constant takes different values for different curves of the original family, and you want an expression for $y^{\prime }$ which is valid for all curves of the family crossed by the orthogonal trajectory you are trying to find. See equations (2.10) to ( 2.12).

$y=k{x}^n$.

(Assume that $n$ is a given number; the different curves of the family have different values of k.)