Answer
$$y=\frac{1}{4}+ce^{-2x}$$
Work Step by Step
Given $$2\frac{dy}{dx} +4y=1$$
Rewriting the equation
$$ \frac{dy}{dx} +2y=\frac{1}{2}$$
Since
\begin{align*}
\mu(x)&=e^{\int 2dx}\\
&=e^{2x }
\end{align*}
Then
\begin{align*}
y\mu(x)&=\int \mu(x) q(x)dx\\
e^{2x }y&=\int \frac{1}{2}e^{2x} dx\\
&= \frac{1}{4}e^{2x}+c
\end{align*}
Hence $$y=\frac{1}{4}+ce^{-2x}$$
