Closed-form exact solutions for periodic motions of a nonlinear oscillator, which contains a quadratic mixed-parity restoring force, are derived analytically. This oscillator is characterised by two real parameters, which are the coefficients of the linear and the quadratic terms, and has single-well potential. All possible combinations of positive and negative values of these coefficients providing periodic motions are considered, and two families of exact solutions are obtained in terms of Jacobi elliptic functions. The periods are given in terms of the complete elliptic integral of the first kind, the behaviour of these periods as a function of the initial amplitude is analysed, and the exact solutions for certain values of these parameters are plotted.

解析地得出了包含二次混合奇偶校验恢复力的非线性振荡器的周期运动的闭式精确解。该振荡器的特征在于两个实参，分别是线性和二次项的系数，并且具有单阱电势。考虑了提供周期性运动的这些系数的正负值的所有可能组合，并根据雅可比椭圆函数获得了两个精确解系列。根据第一种完整的椭圆积分给出周期，分析这些周期随初始振幅变化的行为，并绘制出这些参数某些值的精确解。