Question

The differential equation of the family of curves y² = 4a(x + a) is ______.

Options

• `y^2 - 4 ("d"y)/("d"x)(x + ("d"y)/("d"x))`

• `2y ("d"y)/("d"x)` = 4a

• `y ("d"^2y)/("d"x^2) + (("d"y)/("d"x))^2` = 0

• `2x ("d"y)/("d"x) + y(("d"y)/("d"x))^2 - y`

Answer 1

The differential equation of the family of curves y² = 4a(x + a) is `2x ("d"y)/("d"x) + y(("d"y)/("d"x))^2 - y`.

Explanation:

The given equation of family of curves is y² = 4a(x + a) 

⇒ y² = 4ax + 4a²   .......(1)

Differentiating both sides, w.r.t. x, we get

`2y * ("d"y)/("d"x)` = 4a

⇒ `y * ("d"y)/("d"x)` = 2a

⇒ `y/2 ("d"y)/("d"x)` = a

Now, putting the value of a in equation (1) we get

`y^2 = 4x(y/2 ("d"y)/("d"x)) + 4(y/2 * ("d"y)/("d"x))^2`

⇒ `y^2 = 2xy ("d"y)/("d"x) + y^2 (("d"y)/("d"x))^2`

⇒ y = `2x ("d"y)/("d"x) + y(("d"y)/("d"x))^2`

⇒ `2x * ("d"y)/("d"x) + y * (("d"y)/("d"x))^2 - y` = 0