Orthogonal Trajectories/Examples/Concentric Circles

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:

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

By Solution to Separable Differential Equation:

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

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