Let the equation be
y = mx, ...(i)
where m is an arbitrary constant
Differentiating w.r.t. x, we get,
dy/dx = m ...(ii)
Eliminating m, between Eqs (i) and (ii), we get
dy/dx = y/x ...(iii)
whcih is the differential equation of a family of lines.
Now replacing dy/dx by –dx/dy in Eq. (iii), we get,
-dx/dy = x/y
dy/y = - dx/x
Integrating, we get
∫(dy/y) = –∫(dx/x)
log |y| = log |c| – log |x| = log |c/x|
xy = c
which is required orthogonal trajectories.
