The differential equation of the family of parabolas with focus at the origin and the X-axis as axis, is

Options

A. `y((dy)/(dx))^(2)+4x(dy)/(dx)=4y`
B. `-y((dy)/(dx))^(2)=2x(dy)/(dx)-y`
C. `y((dy)/(dx))^(2)+y=2xy(dy)/(dx)`
D. `y((dy)/(dx))^(2)+2xy(dy)/(dx)+y=0`

Explanation

To find the differential equation of the family of parabolas with focus at the origin and the x-axis as the axis of symmetry, we start with the standard form of the parabola.

### Step 1: Write the equation of the parabola
The equation of a parabola with focus at the origin (0, 0) and the x-axis as its axis of symmetry can be expressed as:
\[ y^2 = 4ax \]
where \( a \) is the distance from the focus to the directrix.

### Step 2: Differentiate the equation
To find the differential equation, we need to eliminate the parameter \( a \). First, we differentiate the equation with respect to \( x \):
\[ 2y \frac{dy}{dx} = 4a \]
From this, we can express \( a \) in terms of \( y \) and \( \frac{dy}{dx} \):
\[ a = \frac{2y \frac{dy}{dx}}{4} = \frac{y \frac{dy}{dx}}{2} \]

### Step 3: Substitute \( a \) back into the parabola equation
Now, we substitute this expression for \( a \) back into the original parabola equation:
\[ y^2 = 4a x \]
Substituting \( a \):
\[ y^2 = 4 \left( \frac{y \frac{dy}{dx}}{2} \right) x \]
This simplifies to:
\[ y^2 = 2y \frac{dy}{dx} x \]

### Step 4: Rearrange the equation
To eliminate \( y \), we can rearrange the equation:
\[ y^2 - 2y \frac{dy}{dx} x = 0 \]

### Step 5: Factor the equation
Factoring out \( y \):
\[ y(y - 2x \frac{dy}{dx}) = 0 \]
This gives us two cases: \( y = 0 \) (which corresponds to the x-axis) or:
\[ y - 2x \frac{dy}{dx} = 0 \]

### Step 6: Write the final differential equation
Thus, the differential equation of the family of parabolas is:
\[ y = 2x \frac{dy}{dx} \]