Question

The differential equation of the family of curves x² + y² – 2ay = 0, where a is arbitrary constant, is ______.

Options

• `(x^2 - y^2) ("d"y)/("d"x)` = 2xy

• `2(x^2 + y^2) ("d"y)/("d"x)` = xy

• `2(x^2 - y^2) ("d"y)/("d"x)` = xy

• `(x^2 + y^2) ("d"y)/("d"x)` = 2xy

Answer 1

The differential equation of the family of curves x² + y² – 2ay = 0, where a is arbitrary constant, is `(x^2 - y^2) ("d"y)/("d"x)` = 2xy.

Explanation:

The given equation is x² + y² – 2ay = 0   ......(1)

Differentiating w.r.t. x, we have

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

⇒ `x + y ("d"y)/("d"x) - "a" ("d"y)/("d"x)` = 0

⇒ `x + (y - "a") ("d"y)/("d"x)` = 0

⇒ `(y - "a") ("d"y)/("d"x)` = – x

⇒ y – a = `(-x)/(("d"y)/("d"x))`

⇒ a = `y + x/(("d"y)/("d"x))`

⇒ a = `(y * ("d"y)/("d"x) + x)/(("d"y)/("d"x))`

Putting the value of a in equation (1) we get

`x^2 + y^2 - 2y [(y ("d"y)/("d"x) + x)/(("d"y)/("d"x))]` = 0

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

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

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

⇒ `(x^2 - y^2) ("d"y)/("d"x)` = 2xy