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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 · The orthogonal trajectories to a family of curves are the curves that intersect each member of the family at a perfectly perpendicular angle. So given a family of curves, you can change the value of the constant in the equation that models the family, to create a family of many curves, and then sket.
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:
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 $Re(z^4/4) = B$. And finally, $Re(z^4/4) = (x^4-6x^2y^2+y^4)/4$
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 · The curves defined by Equation \ref{eq:4.5.19} are integral curves of Equation \ref{eq:4.5.22}, and the integral curves of \[y'={2xy\over x^2-y^2} \nonumber \] are orthogonal trajectories of the family Equation \ref{eq:4.5.19}.
Orthogonal trajectories occur in various branches of physics. In an electrostatic field, the lines of force are orthogonal to the lines of constant potential. The streamlines in aerodynamics are orthogonal trajectories of the velocity-equipotential curves.
