Question

Problem 8 [15 points: A model for the population P(t) in a suburb of a city (in thousands) is given by the initial-value problem dP dt = P(10-P), P(0)-5, where t is measured in months (a) Solve this IVP for P(t)7. (b) What is the limiting value of the population [3] -half of this limiting value pr

Answer 1

dp/dt = p (10 - p) p (0) = 5 dp/p (10 - p) dt Integrating both sides 1/10 integral (1/p + 1/10 p) do = integral dp 1/10 [log (p) - log (10 - p)] = t + c 1/10 log (p/10 - p] = t + c when t = 0 p = 5 1/10 log [5/10 - 5] = 0 + c 1/10 log (1) = c Rightarrow c = 0 we get 1/10 log [p/10 - p] t log [p/10 -p] = 10 t p/10 - p = e^10 t we get 1/10 log [p/10 - p] = t log [p/10 - p] = 10 t p/10 - p = e^10 t p = (10 - p) e^10 t p + p e^10 t = 10 e^10 t p (1 = e^10 t) = 10 e^10 t \r\n p(t) = 10 e^10 t/1 + e^10 t lt_t Rightarrow infinity p(t) = lt_t Rightarrow infinity 10 e^10 t/1 + e^10 t + 1 = 10/p + 1 = 10