Example 1 :
Find the orthogonal trajectories of family of circles , where g is the parameter.
Solution :
The given family of curves
…(1)
Differentiate both sides w.r.t. x,
…(2)
Put the value of g in (1)
…(3)
which is the differential equation of given family.
Now replace by
in (3)
So
put
put in (4) …(5)
which is linear differential equation
Now
Multiply both sides (5) with IF, it becomes.
where d is the parameter.
Example 2 :
Show that one parameter family of curves are self orthogonal.
Solution :
We are given …(1)
Differentiate …(2)
Put value of c in (1)
(3) gives the differential equation of given family
Now replace y’ by in (3), we get the differential equation of orthogonal trajectory.
Which is same as in equation (3)
So, differential equation of given curve and differential equation of its orthogonal trajectaries are same. So, the family of curves is self orthogonal.
Example 3 :
Find the orthogonal trajectories of the family of curves , where
is a parameter.
Solution :
We are given
Differentiate both sides w.r.t x
…(1)
On solving
Now on substituting values of and
in (i)
which is the differential equation is given family. …(2)
Now replace with
to obtain differential equation of orthogonal trajectory
Differential equation (2) and (3) are same, which gives the differential equation of family. It’s orthogonal trajectories are same. So the family of curves are self orthogonal.
Example 4 :
Show that the families of curves given by the equation and
intersect orthogonally.
Solution :
Here we have to show that the family of orthogonal trajectory of the family of curves
We have …(1)
On taking logrithm both sides
Differentiate both sides w.r.t
…(2)
which is free from parameter. So, it is the differential equation of given family.
Now replace with
in (2)
On integrating
which is the required orthogonal trajectory.