`=>`(a) If the given family `F_1` of curves depends on only one parameter then it is represented by an equation of the form
`color {red} {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.,
`color {red} {g (x, y, y′, a) = 0` ... (2)
● The required differential equation is then obtained by eliminating a from equations (1) and (2) as
`color {red} {F(x, y, y′) = 0` ... (3)
`=>` (b) If the given family F2 of curves depends on the parameters a, b (say) then it is represented by an equation of the from
`color {red} {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.,
`color {red} {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
`color {red} {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
`color {red} {F (x, y, y′, y″) = 0` ... (7)
`"Key Concept : "` The order of a differential equation representing a family of curves is same as the number of arbitrary constants present in the equation corresponding to the family of curves.
`=>`(a) If the given family `F_1` of curves depends on only one parameter then it is represented by an equation of the form
`color {red} {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.,
`color {red} {g (x, y, y′, a) = 0` ... (2)
● The required differential equation is then obtained by eliminating a from equations (1) and (2) as
`color {red} {F(x, y, y′) = 0` ... (3)
`=>` (b) If the given family F2 of curves depends on the parameters a, b (say) then it is represented by an equation of the from
`color {red} {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.,
`color {red} {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
`color {red} {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
`color {red} {F (x, y, y′, y″) = 0` ... (7)
`"Key Concept : "` The order of a differential equation representing a family of curves is same as the number of arbitrary constants present in the equation corresponding to the family of curves.