The motion of a mass-spring system with damping is governed by $y''(t) + by'(t) + 36y(t) = 0$; $y(0) = 4$, and $y'(0) = 0$. Find the equation of motion and sketch its graph for $b = 0, 8, 12$, and $20$.
What is the equation of motion for $b = 0$?
$y(t) = 4 \cos 6t$
(Type an exact answer, using radicals as needed.)
What is the equation of motion for $b = 8$?
$y(t) = e^{-4t} \left[ 4 \cos \left( 2\sqrt{5}t \right) + \frac{8\sqrt{5}}{5} \sin \left( 2\sqrt{5}t \right) \right]$
(Type an exact answer, using radicals as needed.)
What is the equation of motion for $b = 12$?
$y(t) = (4 + 24t)e^{-6t}$
(Type an exact answer, using radicals as needed.)
What is the equation of motion for $b = 20$?
$y(t) = \frac{9}{2}e^{-2t} - \frac{1}{2}e^{-18t}$
(Type an exact answer, using radicals as needed.)
Choose the correct graph of the solutions. For each of the graphs $b = 0$ is a solid blue line, $b = 8$ is a dotted red line, $b = 12$ is a dashed green line, and $b = 20$ is a dash-dot black line.