Solve the following differential equations :
`x (dy)/(dx)+2y=x^(2)(x !=0)`

Answer

To solve the differential equation \( x \frac{dy}{dx} + 2y = x^2 \) where \( x \neq 0 \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ x \frac{dy}{dx} + 2y = x^2 \]
We can rearrange it to isolate \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} + \frac{2y}{x} = x \]

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

To find the integrating factor \( \mu(x) \), we calculate:
\[ \mu(x) = e^{\int P(x) \, dx} = e^{\int \frac{2}{x} \, dx} = e^{2 \ln |x|} = |x|^2 \]
Since \( x \neq 0 \), we can drop the absolute value:
\[ \mu(x) = x^2 \]

### Step 3: Multiply through by the integrating factor
Now we multiply the entire differential equation by the integrating factor \( x^2 \):
\[ x^2 \frac{dy}{dx} + 2xy = x^3 \]

### Step 4: Recognize the left-hand side as a derivative
The left-hand side can be expressed as the derivative of a product:
\[ \frac{d}{dx}(x^2 y) = x^3 \]

### Step 5: Integrate both sides
Now we integrate both sides with respect to \( x \):
\[ \int \frac{d}{dx}(x^2 y) \, dx = \int x^3 \, dx \]
This gives us:
\[ x^2 y = \frac{x^4}{4} + C \]
where \( C \) is the constant of integration.

### Step 6: Solve for \( y \)
To find \( y \), we divide both sides by \( x^2 \):
\[ y = \frac{x^4}{4x^2} + \frac{C}{x^2} = \frac{x^2}{4} + \frac{C}{x^2} \]

### Final Solution
Thus, the general solution to the differential equation is:
\[ y = \frac{x^2}{4} + \frac{C}{x^2} \]

---