Answer
See below
Work Step by Step
Given: $\frac{d^2y}{dt^2}+3\frac{dy}{dt}+2y=0$
The motion is governed by the differential equation
$\frac{d^2y}{dt^2}+\frac{c}{m}\frac{dy}{dx}+\frac{k}{m}y=0$
with $\frac{c}{m}=3\\
\frac{k}{m}=2$
then $(\frac{c}{2m})^2=\frac{9}{4}$
We can see $(\frac{c}{2m})^2 \gt \frac{k}{m}$
Thus, the oscillator is criticially overdamped.
The complementary function for the given equation is:
$y(t)=c_1e^{-2t}+c_2e^{-t}$
Since $y(0)=1,y'(0)=3$
We have $c_2=2,c_1=-1$
The general solution is $y(t)=-e^{-2t}+2e^{-t}$
