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30 cze 2023 · If you know the equation of one family of curves, you can find the orthogonal trajectories by finding the solution to the differential equation that describes the orthogonal family. This typically involves taking the derivative of the given equation, negating it, and finding the reciprocal to determine the slope of the orthogonal trajectories.
20 sty 2019 · The orthogonal trajectories to a family of curves are the curves that intersect each member of the family at a perfectly perpendicular angle. So given a family of curves, you can change the value of the constant in the equation that models the family, to create a family of many curves, and then sket.
Below we describe an easier algorithm for finding orthogonal trajectories \(f\left( {x,y} \right) = C\) of the given family of curves \(g\left( {x,y} \right) = C\) using only ordinary differential equations. The algorithm includes the following steps:
So $$3x^2y - y^3 = (3y^2x -x^3)\frac{dy}{dx}, $$ and hence $$ \frac{dy}{dx} = \frac{3x^2y - y^3}{3y^2x - x^3}. $$ Now, the slope of the other family of curves will be the negative inverse, so for the orthogonal trajectory, we have $$\frac{dy}{dx} = -\frac{3y^2x-x^3}{3x^2y-y^3}, $$ which we can write as $$-(3y^2x-x^3)dx = (3x^2y-y^3)dy. $$
29 sty 2018 · Our method of finding the orthogonal trajectories of a given family of curves is therefore as follows: first, find the differential equation of the family; next, replace d y= d x by d x= d y to obtain the differential equation of the orthogonal trajectories; and finally, solve this new
In mathematics, an orthogonal trajectory is a curve which intersects any curve of a given pencil of (planar) curves orthogonally. For example, the orthogonal trajectories of a pencil of concentric circles are the lines through their common center (see diagram).
An orthogonal trajectory of a family of curves is a curve that intersects each curve of the family orthogonally—that is, at right angles. ORTHOGONAL TRAJECTORY Each member of the family y = mx of straight lines through the origin is an orthogonal trajectory of the family x2 + y2 = r2 of concentric circles with center the origin.
