Answer
$$y=2-e^{t }$$
Work Step by Step
Given $$ \frac{dy}{dt} + y =2,\ \ \ \ \ y(0)=1$$
Since
\begin{align*}
\mu(t)&=e^{\int dt}\\
&=e^{t }
\end{align*}
Then
\begin{align*}
y\mu(t)&=\int \mu(t) q(t)dt\\
e^{t } y&=\int 2 e^{t } dt\\
&= 2 e^{t } +c
\end{align*}
Since $y(0)=1,$ then $c=-1$
Hence $$y=2-e^{t }$$
