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