Thwarted in his attempt to generate infinitely many minions (and thereby destroy the world), Dr. Nefario is building an underwater harmonic oscillator deep in the Mariana trench. Dr. Nefario hopes to harness energy from temperature differentials in ocean water to power a giant laser beam (which he will use to destroy the world). Dr. Nefario has hired you as a summer intern to calibrate the harmonic oscillator. You hang a weight of mass m= 1 from the harmonic oscillator, and set it in motion. Remembering what you learned in class you know that the height of the mass satisfies the differential equation: y" + by' + ky = 0 for constants b, k, which are unknown to you and Dr. Nefario. However, from direct observation you know
(i) The solution y(t) attains a local marimum every two seconds. That is, if y(to) is a local maximum, then so is y(to + 2).
(ii) The height at every local maximum is exactly half as large as the preceding local maximum. That is, if y(to) is a local maximum, then
y(to + 2)/y(to) = 1/2 Determine the constants b and k.