The diff. equation of the family of parabolas `y^(2)=4ax` is ……….

Answer

To find the differential equation of the family of parabolas 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 parabola:
\[ y^2 = 4ax \]
Now, we differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(y^2) = \frac{d}{dx}(4ax) \]
Using the chain rule on the left side and the product rule on the right side, we get:
\[ 2y \frac{dy}{dx} = 4a \]

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

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

### Step 4: Rearrange the equation
Next, we can rearrange the equation to isolate terms:
\[ y^2 - 2y \frac{dy}{dx} x = 0 \]

### Step 5: Factor out common terms
We can factor out \( y \) from the left-hand side:
\[ y(y - 2x \frac{dy}{dx}) = 0 \]
Since we are looking for a non-trivial solution (where \( y \neq 0 \)), we set the other factor to zero:
\[ y - 2x \frac{dy}{dx} = 0 \]

### Final Differential Equation
Thus, the differential equation of the family of parabolas is:
\[ 2x \frac{dy}{dx} - y = 0 \]