3.7.2 Number 1
In each case, you want to solve the equation
4 y'' + c y' + 48y = 0
with the initial conditions
y(0) = 0.5 and y'(0)=0. The characteristic
equation has roots
The system is critically damped when
c>27.7, and underdamped (leading to oscillation) when c<27.7.
Thus, in all the given cases, the system is overdamped, and will not
I'll omit detailed calculations, but for each case, the above formula gives
two distinct real roots, say
and the general
solution has the form
Just plug in the initial conditions and solve for c1 and c2.