Form a differential equation of the family of the curves `y^(2)=4ax`

Answer

To form the differential equation of the family of curves given by the equation \( y^2 = 4ax \), we will follow these steps:

### Step 1: Differentiate the given equation
We start with the equation of the curve:
\[ y^2 = 4ax \]
We will differentiate both sides with respect to \( x \).

### Step 2: Apply differentiation
Using implicit differentiation, we differentiate the left side and the right side:
\[ \frac{d}{dx}(y^2) = \frac{d}{dx}(4ax) \]
This gives us:
\[ 2y \frac{dy}{dx} = 4a \]

### Step 3: Solve for \( a \)
From the equation \( 2y \frac{dy}{dx} = 4a \), we can express \( a \) in terms of \( y \) and \( \frac{dy}{dx} \):
\[ a = \frac{1}{2} y \frac{dy}{dx} \]

### Step 4: Substitute \( a \) back into the original equation
Now, we substitute the expression for \( a \) back into the original equation \( y^2 = 4ax \):
\[ y^2 = 4 \left( x \cdot \frac{1}{2} y \frac{dy}{dx} \right) \]
This simplifies to:
\[ y^2 = 2xy \frac{dy}{dx} \]

### Step 5: Rearrange the equation
Now, we rearrange the equation to form the differential equation:
\[ y^2 - 2xy \frac{dy}{dx} = 0 \]

This is the required differential equation for the family of curves \( y^2 = 4ax \).

### Final Answer
The differential equation is:
\[ y^2 - 2xy \frac{dy}{dx} = 0 \]

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