Orthogonal Trajectories/Examples/Parabolas Tangent to X Axis
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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 $1 \ \text b$