Solve the following differential equations :
`(dy)/(dx)+y/x=x^(2)`

Answer

To solve the differential equation

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

we can follow these steps:

### Step 1: Identify the standard form
The given equation can be compared with the standard linear form of a first-order differential equation:

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

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 we are typically interested in positive \( x \) for this context, we can take \( \mu(x) = x \).

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

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

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

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

### 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.

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

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

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

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