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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.
Since holomorphic functions preserve angles (except when the derivative vanish, here it is at $z=0$), the curves that are always orthogonal to those, are the pullback of the vertical lines by $z \mapsto z^4/4$, i.e. the curves $Re(z^4/4) = B$.
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:
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.
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.
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