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solutions to DE system are periodic

Source: Putnam 1982 A4

September 20, 2021
functiondedifferential equationsystem of differential equatio

Problem Statement

Assume that the system of differential equations y′=−z3y'=-z^3, z′=y3z'=y^3 with the initial conditions y(0)=1y(0)=1, z(0)=0z(0)=0 has a unique solution y=f(x)y=f(x), z=g(x)z=g(x) defined for real xx. Prove that there exists a positive constant LL such that for all real xx, f(x+L)=f(x),g(x+L)=g(x).f(x+L)=f(x),\enspace g(x+L)=g(x).
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