Calculation of Approximations to the Periodic Solutions for the Cubic- Root Oscillator

The cubic-root oscillator (CRO) is modeled by the following second-order, nonlinear differential equation (*) $\begin{array}{l} \ddot {x}+x^{\frac{1}{3}}=0,x(0)=A,\dot {x}(0)=0. \\ \\ \end{array}$ First, we show that all solutions to Eq. (*) are periodic. Second, we calculate the exact value of for the period T(A). Third, two techniques are used to calculate approximations for the periodic solutions; these techniques are the methods of harmonic balance and iteration. Generalization of this methodology to other ``truly nonlinear (TNL)'' oscillators will also be discussed.