The paper is devoted to a study of the period function

$T(x,y)$, which associates to every point

$(x,y)$ from a neighborhood of the center

$O$ of the equation

${x}^{\text{'}\text{'}}+f\left(x\right){{x}^{\text{'}}}^{2}+g\left(x\right)=0$ the corresponding period

$T$. The function

$T$ has a strong relationship to the existence and uniqueness of the solutions of some boundary value problem. The author considers some classes of planar systems equivalent to such equation. The article contains a sufficient condition for the monotonicity of

$T$, or for the isochronicity of

$O$, which is also necessary, when

$f$ and

$g$ are odd and analytic.