ps04_solns - 18.03 PS4a PS4b Solutions-Spring 09 Part A 1a...

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18.03 PS4a & PS4b Solutions—Spring 09 Part A 1a, 1b. The complex-valued solution satisfies x 0 + 3 x = e 2 it , so x = ce 2 it for some constant c . Then x 0 + 3 x = 2 ice 2 it + 3 ce 2 it = e 2 it = 2 ic + 3 c = 1 = c = 1 / (3 + 2 i ). The real-valued solution is Re 1 3 + 2 i e 2 it = Re 3 - 2 i 13 (cos 2 t + i sin 2 t ) = 3 13 cos 2 t + 2 13 sin 2 t or alternatively, using that 3 + 2 i = 13 e where 0 < φ < π/ 2 and tan φ = 2 / 3 , Re 1 13 e 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 0 = - 0 . 25 + y - y 2 = - ( y - 0 . 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.
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