Analytical Approximate Solutions for the Cubic-Quintic Duffing Oscillator in Terms of Elementary Functions

Table 1

Percentage errors for comparison of the approximate expression for the complete elliptic integral of the first kind, 𝐾_𝑎(𝑚) (3.7), with the exact one, 𝐾(𝑚).

𝑚	𝐾𝑎(𝑚)	𝐾(𝑚)	(% Error)
0	1.570796	1.570796	0
0.1	1.612441	1.612441	0.00000075
0.2	1.659623	1.659624	0.000015
0.3	1.713888	1.713889	0.00010
0.4	1.777512	1.777519	0.00041
0.5	1.854049	1.854075	0.0014
0.6	1.949485	1.949568	0.0042
2/3	2.028783	2.028959	0.0087