On the analytical approximation of the quadratic non-linear oscillator by modified extended iteration method

Adopted approximate frequencies of x¨+x2=0 \ddot x + {x^2} = 0 _

Exact frequency Ωe=0.914681A {\Omega _e} = 0.914681\sqrt A
Amplitude A
First approximate frequencies, Ω0	0.921318A 0.921318\sqrt A	Second approximate frequencies, Ω1	0.914752A 0.914752\sqrt A	Third approximate frequencies, Ω2	0.91467A 0.91467\sqrt A
Error (%)	0.73	Error (%)	0.0078	Error (%)	0.0012

Comparison of the approximate frequencies with exact frequency Ωe of x¨+x2=0 \ddot x + {x^2} = 0 _

Exact frequency Ωe=0.914681A {\Omega _e} = 0.914681\sqrt A
Amplitude A	First approximate frequencies, Ω0 & Error (%)	Second approximate frequencies, Ω1 & Error (%)	Third approximate frequencies, Ω2 & Error (%)
Mickens and Ramadhani [9]	0.921318A 0.73 0.921318\sqrt A \,0.73	0.914044A 0.70 0.914044\sqrt A 0.70	–
Belendez et al. [14]	0.921318A 0.73 0.921318\sqrt A \,0.73	0.914274A 0.045 0.914274\sqrt A \,0.045	0.914711A 0.0032 0.914711\sqrt A \,0.0032
Hosen M A [13]	0.921318A 0.73 0.921318\sqrt A \,0.73	0.914427A 0.028 0.914427\sqrt A \,0.028	0.914733A 0.0056 0.914733\sqrt A \,0.0056
Haque and Hossain [24]	0.921318A 0.73 0.921318\sqrt A \,0.73	0.915114A 0.047 0.915114\sqrt A \,0.047	0.914705A 0.0026 0.914705\sqrt A \,0.0026
Adopted method	0.921318A 0.73 0.921318\sqrt A \,0.73	0.914752A 0.0078 0.914752\sqrt A \,0.0078	0.91467A 0.0012 0.91467\sqrt A \,0.0012

Comparison of the approximate periods with exact periods Te of x¨+x=−xx˙2 \ddot x + x = - x{\dot x^2} _

A	Te	T Er(%)	TG Er(%)
0.1	6.275334	6.275333 3.43 e−6	6.2753264 1.21 e−4
0.2	6.251809	6.251809 3.61 e−7	6.251690 1.90 e−3
0.5	6.088449	6.088449 3.01 e−6	6.083668 7.85 e−2
1	5.527200	5.527434 4.25 e−3	5.441398 1.55
1.5	4.690247	4.709049 4.7 e−1	4.155936 11.39