Find a differential equation whose general solution is $y=c_{1} e^{-t / 2}+c_{2} e^{-2 t}$.
Then $r_{1}=-\frac{1}{2}, r_{2}=-2$ are two roots of the characteristic equation of the required differential equation.
so the characteristic equation should look like
$$
\begin{aligned}\left(r+\frac{1}{2}\right)(r+2) &=0 \\(2 r+1)(r+2) &=0 \\ 2 r^{2}+5 r+2 &=0 \end{aligned}
$$
which corresponds to the DE
$$
2 y^{\prime \prime}+5 y^{\prime}+2 y=0
$$
