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.
