Formation of differential Equation

Aug 08, 2022, 16:45 IST

Formation of differential Equation

We  know  y² = 4ax  is a  parabola  whose  vertex  is  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) 

From (1)  and (2), y2 = 2yx  y =  2x 

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

(i) 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. 

(ii) The differential equation of a family of curves of two parameter 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. 

Formula for Formation of differential Equation