# Ordinary Differential Equations/Trajectories

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### Orthogonal Trajectory[edit]

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

## Example[edit]

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[edit]

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