Answer
$y=c e^{2 t}-3 e^{t}$
Work Step by Step
$y^{\prime}-2 y=3 e^{t}$
We solve this by integrating factor method. First, find the integrating factor.
$p(t)=-2$
$\mu(t)=e^{\int p(t)} d t$
$\mu(t)=e^{\int -2dt}$
$\mu(t)=e^{-2 t}$
Multiplying both sides with $\mu(t)$
$$
\begin{array}{c}
e^{-2 t}\left(y^{\prime}-2 y\right)=3 \cdot e^{-2 t} \cdot e^{t} \\
y^{\prime} e^{-2 t}-2 e^{-2 t} y=3 e^{-t}
\end{array}
$$
simplifying the left hand side
$$\left(y e^{-2 t}\right)^{\prime}=3 e^{-t}$$
integrating on both sides.
$\int\left(y e^{-2 t}\right)^{\prime} d t=\int 3 e^{-t} d t$
$y e^{-2 t}=-3 e^{-t}+c$
Further dividing both sides by $e^{-2 t}$
$ \quad y=-3 e^{-t+2 t}+c e^{2 t}$
$y=-3 e^{t}+c e^{2 t}$
$y=c e^{2 t}-3 e^{t}$
