MathDB
Miklos Schweitzer 1979_10

Source: system of differential equations

January 28, 2009
functionreal analysisreal analysis unsolved

Problem Statement

Prove that if ai(i=1,2,3,4) a_i(i=1,2,3,4) are positive constants, a2−a4>2 a_2-a_4 > 2, and a1a3−a2>2 a_1a_3-a_2 > 2, then the solution (x(t),y(t)) (x(t),y(t)) of the system of differential equations x˙=a1−a2x+a3xy, \.{x}=a_1-a_2x+a_3xy, y˙=a4x−y−a3xy      (x,y∈R) \.{y}=a_4x-y-a_3xy \;\;\;(x,y \in \mathbb{R}) with the initial conditions x(0)=0,y(0)≥a1 x(0)=0, y(0) \geq a_1 is such that the function x(t) x(t) has exactly one strict local maximum on the interval [0,∞) [0, \infty). L. Pinter, L. Hatvani
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