Chapter 2.4, Problem 32E

To determine

a)

To show:

The curves orthogonal to the family $F\left(x,y\right)=k$ satisfy the differential equation $\frac{\partial F}{\partial y}\left(x,y\right)dx-\frac{\partial F}{\partial x}\left(x,y\right)dy=0$.

To determine

(b)

To show:

The orthogonal trajectories to the family of circles $x^2+y^2=k$ are just straight lines through the origin.

To determine

(c)

To show:

That the orthogonal trajectories to the family of hyperbolas $xy=k$ are the hyperbolas $x^2-y^2=k$.