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20 sty 2019 · In the example below we’ll show how to use calculus to find the orthogonal trajectories, but for now we’ll give away the answer so that we can sketch the family of orthogonal trajectories and see that they are perpendicular to the original family.
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.
let $z = x+iy$ and let $f(x+iy) = x^3y - xy^3$. Expressed in terms of $z$, this is $f(z) = \frac 1 {16i}\left((z+\bar{z})^3(z - \bar z) + (z + \bar z)(z - \bar z)^3\right) = \frac 1{8i}(z^4 - \bar z^4) = \frac 1 4 Im(z^4) = Im(z^4/4)$.
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:
x^2: x^{\msquare} \log_{\msquare} \sqrt{\square} \nthroot[\msquare]{\square} \le \ge \frac{\msquare}{\msquare} \cdot \div: x^{\circ} \pi \left(\square\right)^{'} \frac{d}{dx} \frac{\partial}{\partial x} \int \int_{\msquare}^{\msquare} \lim \sum \infty \theta (f\:\circ\:g) f(x)
