Solve the differential equation:
`x (dy)/(dx)+y=3x^(2)-2`

Answer

To solve the differential equation \( x \frac{dy}{dx} + y = 3x^2 - 2 \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given differential equation:
\[ x \frac{dy}{dx} + y = 3x^2 - 2 \]
To simplify, we divide the entire equation by \( x \):
\[ \frac{dy}{dx} + \frac{1}{x} y = \frac{3x^2 - 2}{x} \]
This simplifies to:
\[ \frac{dy}{dx} + \frac{1}{x} y = 3x - \frac{2}{x} \]

**Hint:** Dividing by \( x \) helps to isolate \( \frac{dy}{dx} \) and makes it easier to identify the linear form.

### Step 2: Identify the integrating factor
The standard form of a linear differential equation is:
\[ \frac{dy}{dx} + P(x) y = Q(x) \]
where \( P(x) = \frac{1}{x} \) and \( Q(x) = 3x - \frac{2}{x} \).

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 \( x > 0 \) in our context, we can simply use \( \mu(x) = x \).

**Hint:** The integrating factor is crucial for solving linear differential equations, as it allows us to combine terms into a single derivative.

### Step 3: Multiply through by the integrating factor
Now we multiply the entire differential equation by the integrating factor \( x \):
\[ x \frac{dy}{dx} + y = x(3x - \frac{2}{x}) \]
This simplifies to:
\[ x \frac{dy}{dx} + y = 3x^2 - 2 \]
Notice that the left-hand side can be expressed as the derivative of a product:
\[ \frac{d}{dx}(xy) = 3x^2 - 2 \]

**Hint:** Recognizing the left side as a derivative of a product simplifies the integration process.

### Step 4: Integrate both sides
Now we integrate both sides with respect to \( x \):
\[ \int \frac{d}{dx}(xy) \, dx = \int (3x^2 - 2) \, dx \]
This gives us:
\[ xy = x^3 - 2x + C \]
where \( C \) is the constant of integration.

**Hint:** When integrating, remember to add the constant of integration, as it represents the family of solutions.

### Step 5: Solve for \( y \)
Now, we can solve for \( y \):
\[ y = \frac{x^3 - 2x + C}{x} \]
This simplifies to:
\[ y = x^2 - 2 + \frac{C}{x} \]

**Hint:** Isolating \( y \) gives us the final form of the solution, which can include a term that approaches zero as \( x \) increases.

### Final Answer
Thus, the general solution to the differential equation is:
\[ y = x^2 - 2 + \frac{C}{x} \]