Question

Calculating orthogonal trajectories?

Answer 1

The equation for the trajectories orthogonal to the family of functions of the form 5 · x² - 2 · y² = C is equal to (1 / 5) · ㏑ x + (1 / 2) · ㏑ y = C.

How to find the equation for the orthogonal trajectories of a given equation

In this problem we have a family of functions in implicit form, that is, a function of the form f(x, y, c) = 0. The equation of a orthogonal trajectory is always perpendicular to a particular form of a equation. First, we determine the first derivative of the given expression:

5 · x² - 2 · y² = C

10 · x - 4 · y · y' = 0

4 · y · y' = 10 · x

y' = (10 · x) / (4 · y)

y' = (5 · x) / (2 · y)

If f(x, y) = (5 · x) / (2 · y), then the differential equation for the orthogonal trajectories related to the family of functions is:

y' = - 1 / f(x, y)

y' = - (2 · y) / (5 · x)

dy / (2 · y) = - dx / (5 · x)

By indefinite integration we get the following solution to the ordinary differential equation:

(1 / 2) · ㏑ y = - (1 / 5) · ㏑ x + C

(1 / 5) · ㏑ x + (1 / 2) · ㏑ y = C

The equation for the trajectories orthogonal to the family of functions of the form 5 · x² - 2 · y² = C is equal to (1 / 5) · ㏑ x + (1 / 2) · ㏑ y = C.

To learn more on orthogonal trajectories: brainly.com/question/20308962

#SPJ1