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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???
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.
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...
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
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 · Step 1. Find a differential equation \[y'=f(x,y) \nonumber \] for the given family. Step 2. Solve the differential equation \[y'=-{1\over f(x,y)} \nonumber \] to find the orthogonal trajectories.
