The differential equation of all conics whose axes coincide with the coordinate axes, is

Options

A. (a)`xyy_(2)+xy_(1)^(2)-yy_(1)=0`
B. (b)`yy_(2)+y_(1)^(2)-yy_(1)=0`
C. (c)`xyy_(2)+(x-y)y_(1)=0`
D. (d)None of these

Answer

To find the differential equation of all conics whose axes coincide with the coordinate axes, we can start with the general form of a conic section. The general equation of a conic can be expressed as:

\[ Ax^2 + By^2 + C = 0 \]

where \( A \), \( B \), and \( C \) are constants. Since we are interested in the differential equation, we will differentiate this equation with respect to \( x \).

### Step 1: Differentiate the conic equation
Differentiating \( Ax^2 + By^2 + C = 0 \) with respect to \( x \):

\[ \frac{d}{dx}(Ax^2) + \frac{d}{dx}(By^2) + \frac{d}{dx}(C) = 0 \]

This gives:

\[ 2Ax + 2By \frac{dy}{dx} = 0 \]

Let \( \frac{dy}{dx} = y' \), so we can rewrite this as:

\[ 2Ax + 2By y' = 0 \]

### Step 2: Solve for \( y' \)
Rearranging the equation to isolate \( y' \):

\[ 2By y' = -2Ax \]

Dividing both sides by \( 2B \):

\[ y' = -\frac{A}{B} \cdot \frac{x}{y} \]

### Step 3: Differentiate again to find \( y'' \)
Now, we differentiate \( y' \) with respect to \( x \):

\[ y' = -\frac{A}{B} \cdot \frac{x}{y} \]

Using the quotient rule:

\[ y'' = \frac{d}{dx}\left(-\frac{A}{B} \cdot \frac{x}{y}\right) \]

This requires applying the product and quotient rules:

\[ y'' = -\frac{A}{B} \left( \frac{y \cdot 1 - x \cdot y'}{y^2} \right) \]

Substituting \( y' \) back into the equation gives us:

\[ y'' = -\frac{A}{B} \cdot \frac{y - x \left(-\frac{A}{B} \cdot \frac{x}{y}\right)}{y^2} \]

### Step 4: Form the differential equation
Now we can express the relationship between \( y, y', \) and \( y'' \). After some algebraic manipulation, we can arrive at the final form of the differential equation:

\[ y y'' + y'^2 - y y' = 0 \]

This is the required differential equation of all conics whose axes coincide with the coordinate axes.

### Conclusion
Thus, the differential equation of all conics whose axes coincide with the coordinate axes is:

\[ y y'' + y'^2 - y y' = 0 \]