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.© BrainMass Inc. brainmass.com December 24, 2021, 8:13 pm ad1c9bdddf
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 ...