Now , differential on both sides of Eq , ( i ) with respect to x and put dy / dx = P .
P = P + x dp / dx + f ’ ( P ) dp / dx = 0
=> [ x + f ’ ( p )] dp / dx = 0
=> dp / dx = 0 => p = C
10 . Orthogonal Trajectory
Any curve , which cuts every member of a given family of curves at right angle , is called an orthogonal trajectory of the family .
Procedure for finding the Orthogonal Trajectory
( i ) Let f ( x , y , c )= 0 be the equation of the given family of curves , where ‘ c ’ is an arbitrary parameter .
( ii ) Differentiate f = 0 , with respect to ‘ x ’ and eliminate 0 , i . e ., from a differential equation .
( iii ) Substitute ( - dx / dy ) for ( dy / dx ) in the above differential equation .
This will give the differential equation of the orthogonal trajectories .
( iv ) By solving this differential equation , we get the required orthogonal trajectories .