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orthogonal trajectories

Hello , how we can find the orthogonal trajectories by mathematica ? I want to find it to x^2 +y^2=c y ,thanks in advance

POSTED BY: Tasneem Smadi
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Thank you so much Sean Clarke .but I want ask you another question if I solve this eq . I can solve it like homogenous eq. or exact eq. I get two equalivant solution how I can prove there equavilant by mathematica ???

POSTED BY: Tasneem Smadi

First ask yourself, how would I do this with a paper and pencil. Then you'll know what steps to take in Mathematica.

The first step is to use Implicit differentiation to find the value of y'. You can do that in Mathematica with:

D[x^2 + y[x]^2 == c y[x], x]

Solve[%, y'[x]]

{y'[x] -> (2 x)/(c - 2 y[x])}

The value orthogonal to this is -1/y'[x].

We can visualize the results:

cPlot = ContourPlot[x^2 + y^2 == 2 y, {x, -2, 2}, {y, -1, 3}, ContourStyle -> Red]

vPlot = VectorPlot[{-2 x, 2 - 2 y}, {x, -2, 2}, {y, -1, 3}]

Show[vPlot, cPlot]

enter image description here

POSTED BY: Sean Clarke
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