The differential equation of all ellipse centered at the origin axis being coordinate axes is:

Options

A. `xy y_2 - xy _(1) ^(2) + y y_1=0`
B. ` y _2 + xy_(1) ^(2) - y y _1=0`
C. `xy y_2 + xy_1^2 + y y _1 =0`
D. none of these

Answer

To find the differential equation of all ellipses centered at the origin with the coordinate axes as their axes, we start with the standard equation of an ellipse:

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

where \(a\) and \(b\) are the semi-major and semi-minor axes respectively, and we assume \(a > b\).

### Step 1: Differentiate the equation with respect to \(x\)

Differentiating both sides with respect to \(x\):

\[ \frac{d}{dx}\left(\frac{x^2}{a^2}\right) + \frac{d}{dx}\left(\frac{y^2}{b^2}\right) = \frac{d}{dx}(1) \]

This gives us:

\[ \frac{2x}{a^2} + \frac{2y}{b^2} \frac{dy}{dx} = 0 \]

### Step 2: Simplify the equation

We can simplify this equation by dividing through by 2:

\[ \frac{x}{a^2} + \frac{y}{b^2} \frac{dy}{dx} = 0 \]

Rearranging gives:

\[ \frac{y}{b^2} \frac{dy}{dx} = -\frac{x}{a^2} \]

### Step 3: Differentiate again

Now we differentiate this equation again with respect to \(x\):

Using the product rule on the left side:

\[ \frac{d}{dx}\left(\frac{y}{b^2}\right) \frac{dy}{dx} + \frac{y}{b^2} \frac{d^2y}{dx^2} = -\frac{1}{a^2} \]

This leads to:

\[ \frac{1}{b^2} \left(\frac{dy}{dx}\right)^2 + \frac{y}{b^2} \frac{d^2y}{dx^2} = -\frac{1}{a^2} \]

### Step 4: Substitute \(b^2/a^2\)

From the original ellipse equation, we can express \(b^2/a^2\) in terms of \(y\) and \(x\):

Using the first derivative equation, we can express \(b^2\) in terms of \(a^2\):

\[ \frac{b^2}{a^2} = -\frac{y \frac{dy}{dx}}{x} \]

### Step 5: Combine the equations

Substituting this back into the differentiated equation gives us:

\[ y \frac{dy}{dx} = x \left(\frac{dy}{dx}\right)^2 + \frac{d^2y}{dx^2} \]

This can be rearranged to:

\[ y \frac{dy}{dx} = x \left(\frac{dy}{dx}\right)^2 + \frac{d^2y}{dx^2} \]

### Final Result

Thus, the differential equation of all ellipses centered at the origin with axes along the coordinate axes is:

\[ y \frac{dy}{dx} = x \left(\frac{dy}{dx}\right)^2 + \frac{d^2y}{dx^2} \]