So now we can write down a differential equation which will determine the family of orthogonal trajectories by negating and reciprocating this last derivative. The following will produce the family of orthogonal trajectories in the form of pure functions. The pure functions should enable us to graph several members of the family of curves. We can now graph some of the members of this family of curves. We will use Table to get a list of graphics objects and then use Show to display them.
Some care needs to be given to the choice of the interval of values for x.
The following may give a warning message about the values of x being outside the domain of one or more of the functions of the family we are studying. So now we will graph some of the members of the original family of curves to then test graphically the orthogonality of the two families of curves. The original family of curves is pretty clearly a family of ellipses centered on the y - axis. We could graph them in several different ways, but will let Mathematica do most of the work for us. So again we get the top halves and the bottom halves of the curves, but we don't have pure functions to work with.