Definition :- A curve which cuts every member of given family of curves according to a given law is called a trajectory of the given family .
---- We shall consider only the case when each trajectory cuts every member of a given family at a constant angle . The trajectory will be called orthogonal , if the constant angle is a right angle . For example , every line through the origin of co-ordinates is an orthogonal trajectory of the family of concentric circle with center at the origin .
How to fined the orthogonal trajectories of the family of curves
where c is a parameter .
Let , phai(x,y,dy/dx)=0
be the differential equation of the family of curves given by
If (dy/dx)=m at a point (x,y) on one of the curves of the system and if another curve cuts that curve at right angle , then m' its slope must be given by the equation
therefore ; m'=-1/m=-dx/dy
Hence at (x,y) on the orthogonal trajectory , these equation must be satisfied
Hence this is the differential equation of the orthogonal system .