# Mathematics - Calculus

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I need to find the orthogonal trajectories of the family of curves, y = 1/(x+c) where k is an arbitrary constant.

So far, I had figured on c = (1/y) - x

m1 = -1/(x^2 + (1/y) - x)

m2 = x^2 + (1/y) - x

I don't know how to figure beyond that. Probably because those were calculated wrong. Please show me how it's done. Thank you.

Â© BrainMass Inc. brainmass.com December 24, 2021, 8:13 pm ad1c9bdddfhttps://brainmass.com/math/basic-calculus/orthogonal-trajectories-family-curves-258003

## SOLUTION This solution is **FREE** courtesy of BrainMass!

y = 1/(x + c)

Differentiating with respect to x, we get dy/dx = -1/(x + c)^2

The differential equation of the orthogonal trajectories is dy/dx = (x + c)^2

dy = (x + c)^2 dx

Integrating on both the sides, we get y = (1/3)(x + c)^3 + k are the orthogonal trajectories.

Members of this family for k = 5 and c = -2, -1, 0, 1 and 2 are shown below ...

https://brainmass.com/math/basic-calculus/orthogonal-trajectories-family-curves-258003