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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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