Step 1: Understanding the Question:
Derive the differential equation by eliminating the arbitrary constant from the family's general equation. Step 2: Key Formula or Approach:
Family of parabolas with vertex $(0,0)$ and axis along positive Y-axis: $x^2 = 4ay$. Step 3: Detailed Explanation:
Equation: $x^2 = 4ay$ -----(1)
Differentiate both sides with respect to $x$:
\[ 2x = 4a \frac{dy}{dx} \] -----(2)
From (1), we have $4a = \frac{x^2}{y}$.
Substitute this into (2):
\[ 2x = \frac{x^2}{y} \frac{dy}{dx} \]
Multiply by $\frac{y}{x}$ (assuming $x \neq 0$):
\[ 2y = x \frac{dy}{dx} \]
\[ x \frac{dy}{dx} - 2y = 0 \] Step 4: Final Answer:
The differential equation is $x \frac{dy}{dx} - 2y = 0$.