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Let f: R -> R be a differentiable function such that y = f(x) satisfies the differential equation dy/dx = (2019 + 2020*y)*(2019 - 2020*y). If lambda = lim_(x -> +infinity) f(-x), find [lambda], where [.] denotes the greatest integer function.

1. (A) 1
2. (B) 2
3. (C) 3
4. (D) 4

Correct answer: (A) 1

Solution

The ODE has two equilibria: y = +2019/2020 (stable as x increases) and y = -2019/2020 (unstable). As x -> -inf (i.e., t -> -inf where we go backward in the ODE), the solution approaches the unstable equilibrium -2019/2020. Thus lambda = -2019/2020 and [lambda] = [-0.999...] = -1.

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