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Procedure to Form a Differential Equation that Will Represent a Given Family of Curves

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(a) If the given family `F_1` of curves depends on only one parameter then it is represented by an equation of the form
 `F_1` (x, y, a) = 0          ... (1) 
For example, the family of parabolas `y^2` = ax can be represented by an equation of the form f (x, y, a) : `y^2` = ax.
Differentiating equation (1) with respect to x, we get an equation involving y′, y, x, and a, i.e.,
g (x, y, y′, a) = 0            ... (2) 
The required differential equation is then obtained by eliminating a from equations (1) and (2) as 
F(x, y, y′) = 0                 ...(3) 

b) If the given family `F_2` of curves depends on the parameters a, b (say) then it is represented by an equation of the from 
`F_2` (x, y, a, b) = 0     ... (4) 
Differentiating equation (4) with respect to x, we get an equation 
involving y′, x, y,  a, b, i.e., 
g (x, y, y′, a, b) = 0      ... (5) 

But it is not possible to eliminate two parameters a and b from the two equations and so, we need a third equation. This equation is obtained by differentiating equation (5), with respect to x, to obtain a relation of the form      h (x, y, y′, y″, a, b) = 0     ... (6)

The required differential equation is then obtained by eliminating a and b from equations (4), (5) and (6) as
F (x, y, y′, y″) = 0       ... (7) 

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