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