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(a) For what values of k does the function y = cos(kt) satisfy the differential equation 16y” = k: —81y? (Enter your answers as a comma-separated list.) (b) For those values of k, verify that every member of the family of functions y = A sin(kt) + B cos(kt) is also a solution.
y = A sin(kt) + B cos(kt) s y' = Ak cos(kt) — Bk sin(kt) => y" = —Ak2 sin(kt) — Bk2 cos(kt). The given differential equation 16y” = —81y is equivalent to 16y“ + 81y = . Thus, LHS = 16y” + 81y= 16(—Ak2 sin(kt) — Bkz cos(kt)) + 81( ) = —16Ak2 5mm) — 163k2 cus(kt) + ( = (81 — 16k2)A sin(kt) + ( )cos(kt) since k2 = )sin(kt) + 815 cos(kt)

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5. 0414.28 points SCalcET8 9.1.009.Ml. A population is modeled by the differential equation dP _ P( p )
— — 1. 1 - .
dt 3 4400 (a) For what values of P is the population increasing? (Enter your answer in interval notation.) PE (b) For what values of P is the population decreasing? (Enter your answer in interval notation.) PE (c) What are the equilibrium solutions? (Enter your answers as a comma-separated list.)

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