#### notes

(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)