Ordinary Differential Equations/Trajectories

Orthogonal Trajectory

Let A be a family of curves. Then B is an orthogonal trajectory of A if every member of B(also a family of curves) cuts every member of A at right angle.It is important to note that we are not insisting that B should intersect every member of A but if they intersect, the angle between their tangents, at every point of intersection, is ${\pi }/2$

Example

Every straight line passing through origin is a normal to every circle having origin as the center. Hence they are orthogonal trajectories of each other.

Steps to find orthogonal trajectory

1. let f(x,y,c)=0 be the equation of the family of curves, where c is an arbitrary constant.
2. Differentiate the given equation with respect to x and then eliminate c.
3. replace $dy/dx$ by $-dx/dy$
4. Solve the obtained differential equation. You will get the required orthogonal trajectory.