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.