Question

What is the formation of the differential equation?

Answer 1

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Here is your answer________

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 y^2 = 4ax      . .  (1)

2y dy/dx = 4a     . .  (2)

From (1) and (2),

y2 = 2y x dy/ 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.

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