Orthogonal Trajectories/Examples/Concentric Circles
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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