Answer
$$y=\frac{1}{2}x+\frac{3}{2x} $$
Work Step by Step
Given $$x \frac{dy}{dx} + y =x,\ \ \ \ \ y(1)=2$$
Rewriting the equation
$$ \frac{dy}{dx} +\frac{1}{x} y=1$$
Since
\begin{align*}
\mu(x)&=e^{\int \frac{1}{x} dx}\\
&=e^{\ln x } \\
&=x
\end{align*}
Then
\begin{align*}
y\mu(x)&=\int \mu(x) q(x)dx\\
x y&=\int x dx\\
&=\frac{1}{2}x^2+c
\end{align*}
since $y(1)=2,$ then $c= \frac{3}{2}$
Hence $$y=\frac{1}{2}x+\frac{3}{2x} $$
