Step 1: Understanding the Concept:
To form a differential equation from a given family of curves with one arbitrary constant, we need to differentiate the given equation with respect to the independent variable ($x$) once.
The goal is to obtain an equation that does not contain the arbitrary constant '$c$'. Step 2: Key Formula or Approach:
Given equation: $x^2y = 4e^x + c$.
Differentiate both sides with respect to $x$ using the product rule on the left side.
The constant '$c$' will disappear upon differentiation since $\frac{d}{dx}(c) = 0$. Step 3: Detailed Explanation:
The given family of curves is:
\[ x^2y = 4e^x + c \]
Differentiate both sides with respect to $x$:
\[ \frac{d}{dx}(x^2 \cdot y) = \frac{d}{dx}(4e^x + c) \]
Apply the product rule ($u'v + uv'$) to the left side:
\[ \left( \frac{d}{dx}(x^2) \right) \cdot y + x^2 \cdot \left( \frac{d}{dx}(y) \right) = \frac{d}{dx}(4e^x) + \frac{d}{dx}(c) \]
\[ (2x) \cdot y + x^2 \cdot \frac{dy}{dx} = 4e^x + 0 \]
Rearrange the terms to match the format of the given options:
\[ x^2 \frac{dy}{dx} + 2xy = 4e^x \]
Bring $4e^x$ to the left side:
\[ x^2 \frac{dy}{dx} + 2xy - 4e^x = 0 \]
This equation matches option (D). Step 4: Final Answer:
The differential equation is $x^2 \frac{dy}{dx} + 2xy - 4e^x = 0$.