Let F be the family of ellipse whose centre is the origin and major axis is the y-axis. Then the differential equation of family F is

Options

A. `(d^(2)y)/(dx^(2))+(dy)/(dx)(x(dy)/(dx)-y)=0`
B. `xy(d^(2)y)/(dx^(2))-(dy)/(dx)(x(dy)/(dx)-y)=0`
C. `xy(d^(2)y)/(dx^(2))+(dy)/(dx)(x(dy)/(dx)-y)=0`
D. `(d^(2)y)/(dx^(2))-(dy)/(dx)(x(dy)/(dx)-y)=0`

Explanation

To find the differential equation of the family of ellipses whose center is at the origin and whose major axis is along the y-axis, we can start with the standard form of such an ellipse.

### Step 1: Write the standard equation of the ellipse.
The standard form of the ellipse with the center at the origin and the major axis along the y-axis is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
where \( a \) and \( b \) are the semi-major and semi-minor axes, respectively.

### Step 2: Rearrange the equation.
Rearranging the equation gives:
\[ b^2 y^2 = a^2 (1 - x^2) \]

### Step 3: Differentiate the equation with respect to \( x \).
Now, we differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(b^2 y^2) = \frac{d}{dx}(a^2 (1 - x^2)) \]
Using the product rule on the left side:
\[ 2b^2 y \frac{dy}{dx} = -2a^2 x \]

### Step 4: Solve for \( \frac{dy}{dx} \).
Rearranging gives:
\[ b^2 y \frac{dy}{dx} = -a^2 x \]
\[ \frac{dy}{dx} = -\frac{a^2 x}{b^2 y} \]

### Step 5: Differentiate again to eliminate parameters.
Next, we differentiate \( \frac{dy}{dx} \) with respect to \( x \) to find \( \frac{d^2y}{dx^2} \):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(-\frac{a^2 x}{b^2 y}\right) \]
Using the quotient rule:
\[ \frac{d^2y}{dx^2} = -\frac{b^2 \cdot a^2 \cdot 1 \cdot y - a^2 x \cdot b^2 \frac{dy}{dx}}{(b^2 y)^2} \]

### Step 6: Substitute \( \frac{dy}{dx} \) back into the equation.
Substituting \( \frac{dy}{dx} \) into the equation, we can simplify further.

### Step 7: Form the differential equation.
After simplification, we can express the relationship in terms of \( x, y, \frac{dy}{dx}, \) and \( \frac{d^2y}{dx^2} \) to form a differential equation. The resulting equation will be:
\[ x y \frac{d^2y}{dx^2} + y \frac{dy}{dx} - y = 0 \]

### Step 8: Identify the correct form of the differential equation.
The final form of the differential equation for the family of ellipses is:
\[ x y \frac{d^2y}{dx^2} + \frac{dy}{dx} - y = 0 \]