Question

The order of the differential equation of a family of curves represented by an equation containing four arbitrary constants, will be

Answer 1

We  know  y2 = 4ax  is a  parabola  whose  vertex is at origin and axis as the x-axis .If a  is a parameter, it  will represent a family of parabola with the  vertex  at (0,  0) and  axis as  y = 0 .

Differentiating y2 = 4ax        . .  (1)

2y dy/dx = 4a     . .  (2)

From (1) and (2), y2 = 2yxdy/ dx & y = 2xdy /dx

This is a differential equation  for all  the  members  of the  family   and  it does  not  contain any  parameter  ( arbitrary constant).

(1) The differential equation of a family of curves of one parameter is a differential equation of the first order, obtained by eliminating the parameter by differentiation.

(2) The differential equation of a family of curves of two parameters is a differential equation of the second order, obtained by eliminating the parameter by differentiating the algebraic equation twice. Similar procedure is used to find differential equation of a family of curves of three or more parameter.

Example: Find the differential equation of the family of curves y = Aex + Be3x for different values of A and B.

Solution:           y = A ex +