Find a family of curves orthogonal to the family of curves given by x^2 + 3y^2 = C.

Question

Find a family of curves orthogonal to the family of curves given by x^2 + 3y^2 = C.

Answer 1

2x + 6yy' = 0; y' = - x/(3y)

So, orthogonal family satisfies ODE: dy/dx = 3y/x; dy/y = 3dx/x; lny = 3lnx + lnC; y = Cx³.