Answer
See below
Work Step by Step
Given: $\frac{d^2y}{dt^2}+4\frac{dy}{dt}+7y=0$
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 $\frac{c}{m}=4\\
\frac{k}{m}=7$
then $(\frac{c}{2m})^2=4$
We can see $(\frac{c}{2m})^2\lt \frac{k}{m}$
Thus, the oscillator is criticially underdamped.
The complementary function for the given equation is:
$y(t)=c_1e^{-2t}\sin (\sqrt 3 t)+c_2e^{-2t}\cos (\sqrt 3t)$
Since $y(0)=2,y'(0)=6$
We have $c_2=0,c_1=\frac{10}{\sqrt 3}$
The general solution is $y(t)=\frac{10}{\sqrt 3}e^{-2t}\sin (\sqrt 3 t)+2e^{-2t}\cos (\sqrt 3t)$
