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Mathematics

1 August, 14:48

Let c be a positive number. A differential equation of the form dy dt = ky1 + c where k is a positive constant, is called a doomsday equation because the exponent in the expression ky1 + c is larger than the exponent 1 for natural growth. (a) Determine the solution that satisfies the initial condition y (0) = y0. (b) Show that there is a finite time t = T (doomsday) such that lim t → T - y (t) = [infinity]. y (t) → [infinity] as 1 - cy0ckt →, that is, as t → cy0ck. Define T = cy0ck. Then lim t → T - y (t) = [infinity]. (c) An especially prolific breed of rabbits has the growth term ky1.01. If 2 such rabbits breed initially and the warren has 14 rabbits after three months, then when is doomsday? (Round your answer to two decimal places.) months