Step 1: Understanding the Concept:
This is a first-order linear differential equation.
The standard form is $\frac{dy}{dx} + P(x)y = Q(x)$.
To solve this, we compute an Integrating Factor (I.F.), which allows us to write the left-hand side as a single derivative of a product.
Step 2: Key Formula or Approach:
1. $I.F. = e^{\int P(x) dx}$.
2. The general solution is $y \cdot (I.F.) = \int [Q(x) \cdot (I.F.)] dx + C$.
Step 3: Detailed Explanation:
Given: $\frac{dy}{dx} + \frac{1}{x} y = x^{2}$.
Comparing with the standard form, we identify:
$P(x) = \frac{1}{x}$ and $Q(x) = x^{2}$.
Calculate the Integrating Factor:
\[ I.F. = e^{\int \frac{1}{x} dx} = e^{\ln x} = x \]
Now, multiply the entire differential equation by $x$:
$x \frac{dy}{dx} + y = x^{3}$.
The left side is the derivative of $(yx)$ by the product rule:
$\frac{d}{dx}(yx) = x^{3}$.
Integrating both sides with respect to $x$:
\[ \int \frac{d}{dx}(yx) dx = \int x^{3} dx \]
\[ yx = \frac{x^{4}}{4} + C \]
This matches Option (A).
Step 4: Final Answer:
The general solution is $yx = \frac{x^{4}}{4} + C$.