Orthogonal Trajectories/Examples/Parabolas Tangent to X Axis

Theorem

Consider the one-parameter family of curves of parabolas which are tangent to the $x$-axis at the origin:

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

Its family of orthogonal trajectories is given by the equation:

$x^2 + 2 y^2 = c$

Proof

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

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

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

\(\text {(2)}: \quad\)

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

\(=\)

\(\ds 2 c x\)

\(\ds \leadsto \ \ \)

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

\(=\)

\(\ds -\frac {2 y} x\)

eliminating $c$ between $(1)$ and $(2)$

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

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

So:

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

\(=\)

\(\ds -\dfrac x {2 y}\)

\(\ds \leadsto \ \ \)

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

\(=\)

\(\ds -2 \int y \rd y\)

Solution to Separable Differential Equation

\(\ds \leadsto \ \ \)

\(\ds x^2\)

\(=\)

\(\ds -2 y^2 + c\)

\(\ds \leadsto \ \ \)

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

\(=\)

\(\ds c\)

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 $1 \ \text b$