# Orthogonal Trajectories/Examples/Concentric Circles

Jump to navigation
Jump to search

## Theorem

Consider the one-parameter family of curves:

- $(1): \quad x^2 + y^2 = c$

Its family of orthogonal trajectories is given by the equation:

- $y = c x$

## Proof

We use the technique of formation of ordinary differential equation by elimination.

Differentiating $(1)$ with respect to $x$ gives:

- $2 x + 2 y \dfrac {\d y} {\d x} = 0$

from which:

- $\dfrac {\d y} {\d x} = -\dfrac x y$

Thus from Orthogonal Trajectories of One-Parameter Family of Curves, the family of orthogonal trajectories is given by:

- $\dfrac {\d y} {\d x} = \dfrac y x$

Using the technique of Separation of Variables:

- $\ds \int \frac {\d y} y = \int \frac {\d x} x$

which by Primitive of Reciprocal gives:

- $\ln y = \ln x + \ln c$

or:

- $y = c x$

Hence the result.

$\blacksquare$

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 3$: Families of Curves. Orthogonal Trajectories