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30 cze 2023 · Step 1. Start with the equation of the given family of curves. This equation will typically involve a parameter (usually denoted by C or k) that differentiates one curve in the family from another. Step 2. Differentiate the given equation with respect to x (or y, depending on the function) to find the slope of the curve, dy/dx.
20 sty 2019 · Step-by-step example of how to find the orthogonal trajectories. Example. 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???.???(1)\frac{dy}{dx}=k(1)?????\frac{dy}{dx}=k???
Understanding a family of curves and their trajectories Given a family of curves, like y=kxy=kx, we can choose different values for kk to write the equations of some of the curves in the...
11 mar 2017 · My textbook, George F. Simmons' Differential Equations with Applications and Historical Notes, asks to find the orthogonal trajectory of the family of curves $r = 2Ccos(\theta)$ where C is a parameter.
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 dy=dx by dx=dy to obtain the differential equation of the orthogonal trajectories; and finally, solve this new differential equation.
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:
7 sty 2020 · Orthogonal Trajectories. Two curves C1 and C2 are said to be orthogonal at a point of intersection (x0, y0) if they have perpendicular tangents at (x0, y0). (Figure 4.5.10 ). A curve is said to be an orthogonal trajectory of a given family of curves if it is orthogonal to every curve in the family.
