Question

The differential equation representing the family of ellipse having foci either on the x-axis or on the y-axis centre at the origin and passing through the point (0, 3) is ______.

विकल्प

• xyy' + y² – 9 = 0

• x + yy" = 0

• xyy" + x(y')² – yy' = 0

• xyy' – y² + 9 = 0

Answer 1

The differential equation representing the family of ellipse having foci either on the x-axis or on the y-axis centre at the origin and passing through the point (0, 3) is `underlinebb(xyy^' - y^2 + 9 = 0)`.

Explanation:

`x^2/a^2 + y^2/b^2` = 1

Since, it passes through (0, 3)

∴ `0/a^2 + 9/b^2` = 1

`\implies` b² = 9 

∴  Equation of ellipse becomes;

`x^2/a^2 + y^2/9` = 1  ...(i)

Differential w.r.t. x, we get;

`(2x)/a^2 + (2y)/9 (dy)/(dx)` = 0

`\implies y/x (dy/dx) = (-9)/a^2`

Now, a² = `(-9x)/(yy^')`

∴ From equation (i)

`(-xyy^')/9 + y^2/9` = 1

`\implies` –xxy' + y² = 9

`\implies` xxy' – y² + 9 = 0