Answer
See below
Work Step by Step
Given: $4\frac{d^2y}{dt^2}+12\frac{dy}{dt}+5y=0$
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 $\frac{c}{m}=3\\
\frac{k}{m}=\frac{5}{4}$
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:
$y(t)=c_1e^{-\frac{5}{2}t}+c_2e^{-\frac{t}{2}}$
Since $y(0)=1,y'(0)=-3$
We have $c_2=-\frac{1}{4},c_1=\frac{5}{4}$
The general solution is $y(t)=\frac{5}{4}e^{-\frac{5}{2}t}-\frac{1}{4}e^{-\frac{t}{2}}$
