Question

The differential equation representing the family of parabolas having vertex at origin and axis along positive direction of X-axis is ______.

विकल्प

• `y^2 - 2xy ("d"y)/("d"x)` = 0

• `y^2 + 2xy ("d"y)/("d"x)` = 0

• `y^2 - 2xy ("d"^2y)/("d"x^2)` = 0

• `y^2 + 2xy ("d"^2y)/("d"x^2)` = 0

Answer 1

The differential equation representing the family of parabolas having vertex at origin and axis along positive direction of X-axis is `y^2 - 2xy ("d"y)/("d"x)` = 0.

Explanation:

The differential equation representing the family of parabolas having vertex at origin is

y² = 4ax   ....(i)

Differentiating w.r.t. x, we get

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

⇒ `2y ("d"y)/("d"x) = y^2/x`  ......[From (i)]

⇒ `2yx ("d"y)/("d"x)` = y²

⇒ `y^2 - 2xy ("d"y)/("d"x)` = 0