The solution of the differential equation $$x\frac{dy}{dx} + 2y = x^2$$, $$(x \neq 0)$$ with $$y(1) = 1$$, is:

We begin with the given differential equation

$$x\frac{dy}{dx}+2y=x^{2},\qquad (x\neq 0).$$

To apply the standard linear‐equation method, we first divide every term by $$x$$ so that the derivative term is alone:

$$\frac{dy}{dx}+\frac{2}{x}\,y=x.$$

Now the equation is of the linear form $$\dfrac{dy}{dx}+P(x)\,y=Q(x)$$ where we can read off

$$P(x)=\frac{2}{x},\qquad Q(x)=x.$$

For such an equation we use an integrating factor. The formula for the integrating factor is

$$\text{IF}=e^{\displaystyle\int P(x)\,dx}.$$

Evaluating the integral in the exponent, we have

$$\int P(x)\,dx=\int\frac{2}{x}\,dx=2\ln|x|.$$

Exponentiating gives

$$\text{IF}=e^{2\ln|x|}=|x|^{2}=x^{2}$$

(the absolute value is unnecessary because $$x\neq 0$$, so we simply write $$x^{2}$$).

Next, we multiply every term of the differential equation by this integrating factor $$x^{2}$$:

$$x^{2}\frac{dy}{dx}+2x\,y=x^{3}.$$

The crucial observation is that the left‐hand side is now the derivative of the product $$x^{2}y$$, because we know the product rule tells us

$${d\over dx}(x^{2}y)=x^{2}\frac{dy}{dx}+2x\,y.$$

Thus we can rewrite the entire equation compactly as

$$\frac{d}{dx}\bigl(x^{2}y\bigr)=x^{3}.$$

We now integrate both sides with respect to $$x$$:

$$\int\frac{d}{dx}\bigl(x^{2}y\bigr)\,dx=\int x^{3}\,dx.$$

The left integral simply returns the function inside the derivative, while the right integral is a power integral:

$$x^{2}y=\frac{x^{4}}{4}+C,$$

where $$C$$ is the constant of integration.

Solving for $$y$$ we divide by $$x^{2}$$:

$$y=\frac{x^{4}}{4x^{2}}+\frac{C}{x^{2}}=\frac{x^{2}}{4}+\frac{C}{x^{2}}.$$

To determine the constant $$C$$ we use the initial condition $$y(1)=1$$. Substituting $$x=1$$ and $$y=1$$ into the general solution we get

$$1=\frac{1^{2}}{4}+\frac{C}{1^{2}}=\frac14+C.$$

Hence

$$C=1-\frac14=\frac34.$$

Putting this value back into the expression for $$y$$, we arrive at the particular solution:

$$y=\frac{x^{2}}{4}+\frac{3}{4x^{2}}.$$

Comparing with the options provided, this exactly matches Option C.

Hence, the correct answer is Option C.