Putnam 1982 A4

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