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Unformatted text preview: 18.03 PS4a & PS4b Solutions—Spring 09 Part A 1a, 1b. The complexvalued solution satisfies x +3 x = e 2 it , so x = ce 2 it for some constant c . Then x + 3 x = 2 ice 2 it + 3 ce 2 it = e 2 it = ⇒ 2 ic + 3 c = 1 = ⇒ c = 1 / (3 + 2 i ). The realvalued solution is Re 1 3 + 2 i e 2 it = Re 3 2 i 13 (cos2 t + i sin2 t ) = 3 13 cos2 t + 2 13 sin2 t or alternatively, using that 3 + 2 i = √ 13 e iφ where < φ < π/ 2 and tan φ = 2 / 3 , Re 1 √ 13 e iφ e 2 it = 1 √ 13 Re h e i (2 t φ ) i = 1 √ 13 cos(2 t φ ) 2. See Figure 1. Part B 1a. a < 1 = ⇒ two steady state solutions, one stable and one unstable. a = 1 = ⇒ one steady state solution, semistable. 1 < a < 1 = ⇒ no steady state solutions. a = 1 = ⇒ one steady state solution, semistable. a > 1 = ⇒ two steady state solutions, one stable and one unstable. The Godzillas go extinct when a < 1. The ODE fails to model a population when y < 0, regardless of the value of a . Also a < 0 corresponds to a negative amount of radiation, which seems impossible, but who knows what techniques the secret lab has invented? 1b. The radiation level required is a = 1 . In that case the ODE is y = . 25 + y y 2 = ( y . 5) 2 , so the steady state solution occurs at y = 0 . 5 Godzillas. 1c. If the radiation level is a bit lower than a = 1, the Godzillas go extinct. If the level is a bit higher, the population of Godzillas stabilizes at a level slightly above 0.5.a bit higher, the population of Godzillas stabilizes at a level slightly above 0....
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This note was uploaded on 05/06/2010 for the course 18 18.03 taught by Professor Unknown during the Spring '09 term at MIT.
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