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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 · In the example below we’ll show how to use calculus to find the orthogonal trajectories, but for now we’ll give away the answer so that we can sketch the family of orthogonal trajectories and see that they are perpendicular to the original family.
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 $$-(3y^2x-x^3)dx = (3x^2y-y^3)dy. $$
22 lut 2016 · From the given family of curves, we find a differential equation the curves all satisfy, Letting , we know the orthogonal trajectories are the curves which satisfy a differential equation. Therefore, the orthogonal trajectories are the curves, where is an arbitrary constant. Apostol - Calculus 1. Exercises.
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:
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).
Two curves are orthogonal or perpendicular at a point if their respective tangent lines to the curves at that point are perpendicular. The word orthogonal comes from the Greek ορθη (right) and γωνια (angle); the word trajectory comes from the Latin trajectus (cut across).
