Answer
See below
Work Step by Step
Given: $\frac{d^2y}{dt^2}+5\frac{dy}{dt}+6y=0$
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 $\frac{c}{m}=5\\
\frac{k}{m}=6$
then $(\frac{c}{2m})^2=\frac{25}{4}$
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:
$y(t)=c_1e^{-3t}+c_2e^{-2t}$
Since $y(0)=-1,y'(0)=4$
We have $c_2=1,c_1=-2$
The general solution is $y(t)=2e^{-3t}+e^{-2t}$
