Exact and approximate solutions for the anti-symmetric quadratic truly nonlinear oscillator

The exact solution of the anti-symmetric quadratic truly nonlinear oscillator is derived from the first integral of the nonlinear differential equation which governs the behaviour of this oscillator. This exact solution is expressed as a piecewise function including Jacobi elliptic cosine functions. The Fourier series expansion of the exact solution is also analyzed and its coefficients are computed numerically. We also show that these Fourier coefficients decrease rapidly and, consequently, using just a few of them provides an accurate analytical representation of the exact periodic solution. Some approximate solutions containing only two harmonics as well as a rational harmonic representation are obtained and compared with the exact solution.