Solution of the differential equation
`(dy)/(dx)+(y)/(x)=x^(2)` is

Options

A. `4xy=x^(4)+c`
B. `xy=x^(4)+c`
C. `4xy+x^(4)=c`
D. None

Answer

To solve the differential equation

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

we will follow these steps:

### Step 1: Identify the form of the 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 \( Q(x) = x^2 \).

**Hint:** Recognize the standard form of a linear differential equation.

### 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 this context, we can simply use \( \mu(x) = x \).

**Hint:** The integrating factor is derived from the coefficient of \( y \) in the differential equation.

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

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

**Hint:** Multiplying by the integrating factor helps to simplify the left-hand side into a derivative.

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

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

**Hint:** Recognize that the left-hand side is the derivative of the product of the integrating factor and \( y \).

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

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

This gives us:

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

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

**Hint:** Remember to add the constant of integration after integrating.

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

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

**Hint:** Isolate \( y \) to express it in terms of \( x \).

### Final Solution
Thus, the general solution of the differential equation is:

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

### Summary of Steps
1. Identify the form of the equation.
2. Find the integrating factor.
3. Multiply the equation by the integrating factor.
4. Rewrite the left-hand side as a derivative.
5. Integrate both sides.
6. Solve for \( y \).