- #1

JoeyJoeJoeJr

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Consider y'(t) = y(t)[a(t) - b(t)y(t)] where a,b : (-infinite, +infinite) → (0, +infinity)

and there exists M>0 such that: (1/M) ≤ a(t), b(t) ≤ M, for all t in the reals

Claim: There exists a unique positive solution Phi(t) defined for all t in the reals in which there exists an m>0 such that: (1/m) ≤ Phi(t) ≤ m. (i.e. Phi(t) is bounded and bounded away from zero)

note that a(t) and b(t) are not necessarily periodic.

Prove the claim true or false.

Any help would be tremendously appreciated. I'm practically dying over this one.