Question: find equation of orthogonal trajectories of 1parameter famly of elipse 1/2x^2+y^2=C.
The family of curves is,
1/2x^2 + y^2 = C
differentiate wrt x,
x + 2y.y' = 0
y' = -x/2y
Here y' gives the slope of the current family of curves.
Then -1/y' (i.e. 2y/x) is the slope of the orthogonal family of curves.
A DE describing the orthogonal family of curves is given by
y' = 2y/x
dy/dx = 2y/x
int (1/y) dy = int (2/x) dx
ln(Ky) = 2.ln(x) = ln(x^2)
Ky = x^2
y = Bx^2
The orthogonal family of curves is: y = Bx^2