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20 sty 2019 · Find the orthogonal trajectories to the family of curves. ???y=kx??? We always start by using implicit differentiation to take the derivative of both sides, and then we’ll solve for ???dy/dx???.
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.
Find the orthogonal trajectories of the family of curves given by $x^2 + y^2 + 2Cy =1$.
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:
A curve which intersects each member of a given family of curves at right angles (orthogonally) is called an orthogonal trajectory of the family. In general, when we have two one-parameter families of curves. F (x, y, C) = 0 and G(x, y, K) = 0.
ORTHOGONAL TRAJECTORIES Find the orthogonal trajectories of the family of curves x = ky 2, where k is an arbitrary constant. ORTHOGONAL TRAJECTORIES Example 5 The curves x = ky 2 form a family of parabolas whose axis of symmetry is the x-axis. The first step is to find a single differential equation that is satisfied by all members of the family.
A procedure for finding a family of orthogonal trajectories G(x,y,K) = 0 for a given family of curves F(x,y,C) = 0 is as follows: Step 1. Determine the differential equation for the given family F(x,y,C)=0. Step 2. Replace y0 in that equation by −1/y0; the resulting equation is the differential equation for the family of orthogonal ...
