Orthogonal Trajectories/Examples/Rectangular Hyperbolas

Theorem

Consider the one-parameter family of curves of rectangular hyperbolas:

$(1): \quad x y = c$

Its family of orthogonal trajectories is given by the equation:

$x^2 - y^2 = c$

Proof

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

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

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

\(\ds x \frac {\d y} {\d x} + y\)

\(=\)

\(\ds 0\)

\(\ds \leadsto \ \ \)

\(\ds \frac {\d y} {\d x}\)

\(=\)

\(\ds -\frac y x\)

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

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

So:

\(\ds \frac {\d y} {\d x}\)

\(=\)

\(\ds \frac x y\)

\(\ds \leadsto \ \ \)

\(\ds \int x \rd x\)

\(=\)

\(\ds \int y \rd y\)

Solution to Separable Differential Equation

\(\ds \leadsto \ \ \)

\(\ds x^2\)

\(=\)

\(\ds y^2 + c\)

\(\ds \leadsto \ \ \)

\(\ds x^2 - y^2\)

\(=\)

\(\ds c\)

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: Problem $1 \ \text a$