Orthogonal Trajectories/Examples/Exponential Functions

Theorem

Consider the one-parameter family of curves of graphs of the exponential function:

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

Its family of orthogonal trajectories is given by the equation:

$y^2 = -2 x + c$

Proof

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

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

$\dfrac {\d y} {\d x} = c e^x$

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

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

\(=\)

\(\ds c e^x\)

\(\ds \leadsto \ \ \)

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

\(=\)

\(\ds y\)

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

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

$\dfrac {\d y} {\d x} = -\dfrac 1 y$

So:

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

\(=\)

\(\ds -\dfrac 1 y\)

\(\ds \leadsto \ \ \)

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

\(=\)

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

Solution to Separable Differential Equation

\(\ds \leadsto \ \ \)

\(\ds y^2\)

\(=\)

\(\ds -2 x + c\)

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 $1 \ \text d$