` (dy)/(dx) + (1)/(x) * y = x ^(2)`

Answer

To solve the differential equation

\[ \frac{dy}{dx} + \frac{1}{x} y = x^2, \]

we will follow these steps:

### Step 1: Identify the type of differential equation
This is a first-order linear differential equation of the form

\[ \frac{dy}{dx} + P(x)y = Q(x), \]

where \( P(x) = \frac{1}{x} \) and \( Q(x) = x^2 \).

### Step 2: Find the integrating factor
The integrating factor \( \mu(x) \) is given by

\[ \mu(x) = e^{\int P(x) \, dx} = e^{\int \frac{1}{x} \, dx} = e^{\ln |x|} = |x|. \]

Since \( x \) is positive in our context, we can take \( \mu(x) = x \).

### Step 3: Multiply the entire equation by the integrating factor
Multiply the differential equation by \( x \):

\[ x \frac{dy}{dx} + y = x^3. \]

### Step 4: Rewrite the left-hand side
The left-hand side can be rewritten as the derivative of a product:

\[ \frac{d}{dx}(xy) = x^3. \]

### Step 5: Integrate both sides
Now, integrate both sides with respect to \( x \):

\[ \int \frac{d}{dx}(xy) \, dx = \int x^3 \, dx. \]

This gives:

\[ xy = \frac{x^4}{4} + C, \]

where \( C \) is the constant of integration.

### Step 6: Solve for \( y \)
Now, solve for \( y \):

\[ y = \frac{x^4}{4x} + \frac{C}{x} = \frac{x^3}{4} + \frac{C}{x}. \]

### Final Solution
Thus, the general solution to the differential equation is

\[ y = \frac{x^3}{4} + \frac{C}{x}. \]

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