Chapter 12.5, Problem 17E

Prove that the zero solution for a conservative system

$\frac{d^{2}y}{d{t}^2}+g\left(x\right)=0$,

where $g$ is continuous in an interval containing zero, is stable provided

$g\left(0\right)=0$ $k>0,sg\left(s\right)>0$ $0<\left|s\right|<k$.