The Asymptotic Expansion Method via Symbolic Computation

Table 2

Some coefficients of the asymptotic expansion of the solution to (7.1).

𝑥0=	+1.000000cos(𝑡)
𝑥1=	−0.031250cos(𝑡)	+0.031250cos(3𝑡)	+0.500000sin(𝑡)
	−0.500000𝑡cos(𝑡)	−0.375000𝑡sin(𝑡)
𝑥2=	+0.022461cos(𝑡)	−0.023438cos(3𝑡)	+0.000977cos(5𝑡)
	+0.226563sin(𝑡)	+0.070313sin(3𝑡)	−0.390625𝑡cos(𝑡)
	−0.046875𝑡cos(3𝑡)	−0.031250𝑡sin(𝑡)	−0.035156𝑡sin(3𝑡)
	+0.054688𝑡^2cos(𝑡)	+0.375000𝑡^2sin(𝑡)
𝑥3=	+0.047760cos(𝑡)	−0.046326cos(3𝑡)	−0.001465cos(5𝑡)
	+0.000031cos(7𝑡)	+0.122640sin(𝑡)	−0.005859sin(3𝑡)
	+0.003988sin(5𝑡)	−0.201660𝑡cos(𝑡)	+0.079102𝑡cos(3𝑡)
	−0.002441𝑡cos(5𝑡)	−0.133057𝑡sin(𝑡)	−0.059692𝑡sin(3𝑡)
	−0.001831𝑡sin(5𝑡)	+0.103760𝑡^2cos(𝑡)	+0.015381𝑡^2cos(3𝑡)
	+0.140625𝑡^2sin(𝑡)	+0.070313𝑡^2sin(3𝑡)	+0.084635𝑡^3cos(𝑡)
	−0.194336𝑡^3sin(𝑡)
𝑥4=	+0.031557cos(𝑡)	−0.025802cos(3𝑡)	−0.005688cos(5𝑡)
	−0.000069cos(7𝑡)	+0.000001cos(9𝑡)	+0.314784sin(𝑡)
	+0.001236sin(3𝑡)	−0.003563sin(5𝑡)	+0.000181sin(7𝑡)
	−0.329605𝑡cos(𝑡)	+0.018219𝑡cos(3𝑡)	+0.009552𝑡cos(5𝑡)
	−0.000107𝑡cos(7𝑡)	−0.441519𝑡sin(𝑡)	+0.023666𝑡sin(3𝑡)
	−0.006212𝑡sin(5𝑡)	−0.000080𝑡sin(7𝑡)	+0.340805𝑡^2cos(𝑡)
	−0.113068𝑡^2cos(3𝑡)	+0.001335𝑡^2cos(5𝑡)	+0.323486𝑡^2sin(𝑡)
	−0.007141𝑡^2sin(3𝑡)	+0.005493𝑡^2sin(5𝑡)	−0.113566𝑡^3cos(𝑡)
	+0.031860𝑡^3cos(3𝑡)	−0.168498𝑡^3sin(𝑡)	−0.064362𝑡^3sin(3𝑡)
	−0.081533𝑡^4cos(𝑡)	+0.060547𝑡^4sin(𝑡)
𝑥5=	+0.101090cos(𝑡)	−0.103018cos(3𝑡)	+0.002327cos(5𝑡)
	−0.000396cos(7𝑡)	−0.000003cos(9𝑡)	+0.000000cos(11𝑡)
	+0.567049sin(𝑡)	+0.011443sin(3𝑡)	−0.003304sin(5𝑡)
	−0.000300sin(7𝑡)	+0.000007sin(9𝑡)	−0.630124𝑡cos(𝑡)
	+0.042347𝑡cos(3𝑡)	+0.004305𝑡cos(5𝑡)	+0.000649𝑡cos(7𝑡)
	−0.000004𝑡cos(9𝑡)	−0.861960𝑡sin(𝑡)	−0.136819𝑡sin(3𝑡)
	+0.014968𝑡sin(5𝑡)	−0.000407𝑡sin(7𝑡)	−0.000003𝑡sin(9𝑡)
	+0.769419𝑡^2cos(𝑡)	+0.056821𝑡^2cos(3𝑡)	−0.019625𝑡^2cos(5𝑡)
	+0.000082𝑡^2cos(7𝑡)	+0.653250𝑡^2sin(𝑡)	−0.009047𝑡^2sin(3𝑡)
	−0.002311𝑡^2sin(5𝑡)	+0.000320𝑡^2sin(7𝑡)	−0.271820𝑡^3cos(𝑡)
	+0.077614𝑡^3cos(3𝑡)	+0.003465𝑡^3cos(5𝑡)	−0.512211𝑡^3sin(𝑡)
	+0.066607𝑡^3sin(3𝑡)	−0.007243𝑡^3sin(5𝑡)	+0.084625𝑡^4cos(𝑡)
	−0.054769𝑡^4cos(3𝑡)	+0.110530𝑡^4sin(𝑡)	+0.027557𝑡^4sin(3𝑡)
	+0.046019𝑡^5cos(𝑡)	−0.005018𝑡^5sin(𝑡)
𝑥6=	+0.041699cos(𝑡)	−0.035635cos(3𝑡)	−0.006540cos(5𝑡)
	+0.000498cos(7𝑡)	−0.000022cos(9𝑡)	−0.000000cos(11𝑡)
	+0.000000cos(13𝑡)	+1.753506sin(𝑡)	−0.124680sin(3𝑡)
	+0.000293sin(5𝑡)	−0.000409sin(7𝑡)	−0.000018sin(9𝑡)
	+0.000000sin(11𝑡)	−1.732879𝑡cos(𝑡)	+0.361830𝑡cos(3𝑡)
	−0.007317𝑡cos(5𝑡)	+0.000416𝑡cos(7𝑡)	+0.000035𝑡cos(9𝑡)
	−0.000000𝑡cos(11𝑡)	−1.950362𝑡sin(𝑡)	+0.090323𝑡sin(3𝑡)
	−0.009679𝑡sin(5𝑡)	+0.001762𝑡sin(7𝑡)	−0.000022𝑡sin(9𝑡)
	−0.000000𝑡sin(11𝑡)	+1.753296𝑡^2cos(𝑡)	−0.260570𝑡^2cos(3𝑡)
	+0.014763𝑡^2cos(5𝑡)	−0.001785𝑡^2cos(7𝑡)	+0.000004𝑡^2cos(9𝑡)
	+1.911049𝑡^2sin(𝑡)	+0.270222𝑡^2sin(3𝑡)	−0.022385𝑡^2sin(5𝑡)
	−0.000220𝑡^2sin(7𝑡)	+0.000016𝑡^2sin(9𝑡)	−1.003710𝑡^3cos(𝑡)
	−0.074659𝑡^3cos(3𝑡)	+0.018743𝑡^3cos(5𝑡)	+0.000255𝑡^3cos(7𝑡)
	−1.212436𝑡^3sin(𝑡)	−0.124235𝑡^3sin(3𝑡)	+0.019212𝑡^3sin(5𝑡)
	−0.000552𝑡^3sin(7𝑡)	+0.305366𝑡^4cos(𝑡)	−0.009847𝑡^4cos(3𝑡)
	−0.008565𝑡^4cos(5𝑡)	+0.511480𝑡^4sin(𝑡)	−0.077788𝑡^4sin(3𝑡)
	+0.004177𝑡^4sin(5𝑡)	−0.061540𝑡^5cos(𝑡)	+0.046341𝑡^5cos(3𝑡)
	−0.067702𝑡^5sin(𝑡)	+0.007443𝑡^5sin(3𝑡)	−0.018889𝑡^6cos(𝑡)
	−0.007806𝑡^6sin(𝑡)
𝑥7=	−0.030141cos(𝑡)	+0.026721cos(3𝑡)	+0.003535cos(5𝑡)
	−0.000160cos(7𝑡)	+0.000046cos(9𝑡)	−0.000001cos(11𝑡)
	−0.000000cos(13𝑡)	+0.000000cos(15𝑡)	+5.481497sin(𝑡)
	+0.136811sin(3𝑡)	−0.022934sin(5𝑡)	+0.000391sin(7𝑡)
	−0.000032sin(9𝑡)	−0.000001sin(11𝑡)	+0.000000sin(13𝑡)
	−5.726072𝑡cos(𝑡)	−0.099462𝑡cos(3𝑡)	+0.047957𝑡cos(5𝑡)
	−0.002159𝑡cos(7𝑡)	+0.000029𝑡cos(9𝑡)	+0.000002𝑡cos(11𝑡)
	−0.000000𝑡cos(13𝑡)	−5.240757𝑡sin(𝑡)	+0.347041𝑡sin(3𝑡)
	+0.020306𝑡sin(5𝑡)	−0.000669𝑡sin(7𝑡)	+0.000136𝑡sin(9𝑡)
	−0.000001𝑡sin(11𝑡)	−0.000000𝑡sin(13𝑡)	+4.898568𝑡^2cos(𝑡)
	−0.633927𝑡^2cos(3𝑡)	−0.017751𝑡^2cos(5𝑡)	+0.002044𝑡^2cos(7𝑡)
	−0.000122𝑡^2cos(9𝑡)	+0.000000𝑡^2cos(11𝑡)	+5.910751𝑡^2sin(𝑡)
	−0.423093𝑡^2sin(3𝑡)	+0.042183𝑡^2sin(5𝑡)	−0.003593𝑡^2sin(7𝑡)
	−0.000015𝑡^2sin(9𝑡)	+0.000001𝑡^2sin(11𝑡)	−3.289085𝑡^3cos(𝑡)
	+0.621827𝑡^3cos(3𝑡)	−0.037683𝑡^3cos(5𝑡)	+0.002208𝑡^3cos(7𝑡)
	+0.000015𝑡^3cos(9𝑡)	−3.616263𝑡^3sin(𝑡)	−0.288847𝑡^3sin(3𝑡)
	+0.002661𝑡^3sin(5𝑡)	+0.002305𝑡^3sin(7𝑡)	−0.000034𝑡^3sin(9𝑡)
	+1.238058𝑡^4cos(𝑡)	−0.002983𝑡^4cos(3𝑡)	−0.002782𝑡^4cos(5𝑡)
	−0.000822𝑡^4cos(7𝑡)	+1.664171𝑡^4sin(𝑡)	+0.252574𝑡^4sin(3𝑡)
	−0.030580𝑡^4sin(5𝑡)	+0.000408𝑡^4sin(7𝑡)	−0.311598𝑡^5cos(𝑡)
	−0.037610𝑡^5cos(3𝑡)	+0.009709𝑡^5cos(5𝑡)	−0.443419𝑡^5sin(𝑡)
	+0.045286𝑡^5sin(3𝑡)	+0.002005𝑡^5sin(5𝑡)	+0.041874𝑡^6cos(𝑡)
	−0.023862𝑡^6cos(3𝑡)	+0.045263𝑡^6sin(𝑡)	−0.022910𝑡^6sin(3𝑡)
	+0.005561𝑡^7cos(𝑡)	+0.006555𝑡^7sin(𝑡)
𝑥8=	+0.707697cos(𝑡)	−0.737652cos(3𝑡)	+0.029647cos(5𝑡)
	+0.000299cos(7𝑡)	+0.000006cos(9𝑡)	+0.000003cos(11𝑡)
	−0.000000cos(13𝑡)	−0.000000cos(15𝑡)	+17.859287sin(𝑡)
	+0.537553sin(3𝑡)	+0.020622sin(5𝑡)	−0.001769sin(7𝑡)
	+0.000063sin(9𝑡)	−0.000002sin(11𝑡)	−0.000000sin(13𝑡)
	+0.000000sin(15𝑡)	−19.388600𝑡cos(𝑡)	−0.105739𝑡cos(3𝑡)
	−0.072738𝑡cos(5𝑡)	+0.004113𝑡cos(7𝑡)	−0.000258𝑡cos(9𝑡)
	+0.000002𝑡cos(11𝑡)	+0.000000𝑡cos(13𝑡)	−0.000000𝑡cos(15𝑡)
	−16.767762𝑡sin(𝑡)	−0.906868𝑡sin(3𝑡)	+0.089594𝑡sin(5𝑡)
	+0.000388𝑡sin(7𝑡)	−0.000052𝑡sin(9𝑡)	+0.000008𝑡sin(11𝑡)
	−0.000000𝑡sin(13𝑡)	−0.000000𝑡sin(15𝑡)	+16.120089𝑡^2cos(𝑡)
	+0.421931𝑡^2cos(3𝑡)	−0.092803𝑡^2cos(5𝑡)	+0.001418𝑡^2cos(7𝑡)
	+0.000192𝑡^2cos(9𝑡)	−0.000007𝑡^2cos(11𝑡)	+0.000000𝑡^2cos(13𝑡)
	+19.608050𝑡^2sin(𝑡)	−0.420458𝑡^2sin(3𝑡)	−0.128984𝑡^2sin(5𝑡)
	+0.005901𝑡^2sin(7𝑡)	−0.000351𝑡^2sin(9𝑡)	−0.000001𝑡^2sin(11𝑡)
	+0.000000𝑡^2sin(13𝑡)	−11.042356𝑡^3cos(𝑡)	+0.875871𝑡^3cos(3𝑡)
	+0.101426𝑡^3cos(5𝑡)	−0.006986𝑡^3cos(7𝑡)	+0.000187𝑡^3cos(9𝑡)
	+0.000001𝑡^3cos(11𝑡)	−12.224352𝑡^3sin(𝑡)	+0.801292𝑡^3sin(3𝑡)
	−0.047084𝑡^3sin(5𝑡)	+0.002053𝑡^3sin(7𝑡)	+0.000194𝑡^3sin(9𝑡)
	−0.000002𝑡^3sin(11𝑡)	+4.715957𝑡^4cos(𝑡)	−0.906645𝑡^4cos(3𝑡)
	+0.033974𝑡^4cos(5𝑡)	−0.000385𝑡^4cos(7𝑡)	−0.000061𝑡^4cos(9𝑡)
	+5.809016𝑡^4sin(𝑡)	+0.172731𝑡^4sin(3𝑡)	+0.039748𝑡^4sin(5𝑡)
	−0.004671𝑡^4sin(7𝑡)	+0.000031𝑡^4sin(9𝑡)	−1.468495𝑡^5cos(𝑡)
	+0.132440𝑡^5cos(3𝑡)	−0.017574𝑡^5cos(5𝑡)	+0.001171𝑡^5cos(7𝑡)
	−1.830821𝑡^5sin(𝑡)	−0.280024𝑡^5sin(3𝑡)	+0.025814𝑡^5sin(5𝑡)
	+0.000249𝑡^5sin(7𝑡)	+0.278969𝑡^6cos(𝑡)	+0.046180𝑡^6cos(3𝑡)
	−0.006166𝑡^6cos(5𝑡)	+0.377937𝑡^6sin(𝑡)	−0.001896𝑡^6sin(3𝑡)
	−0.006983𝑡^6sin(5𝑡)	−0.028889𝑡^7cos(𝑡)	+0.004692𝑡^7cos(3𝑡)
	−0.029867𝑡^7sin(𝑡)	+0.021274𝑡^7sin(3𝑡)	−0.000816𝑡^8cos(𝑡)
	−0.003373𝑡^8sin(𝑡)
𝑥9=	+2.175721cos(𝑡)	−2.094928cos(3𝑡)	−0.084502cos(5𝑡)
	+0.003726cos(7𝑡)	−0.000018cos(9𝑡)	+0.000001cos(11𝑡)
	+0.000000cos(13𝑡)	−0.000000cos(15𝑡)	−0.000000cos(17𝑡)
	+64.332742sin(𝑡)	+0.333348sin(3𝑡)	+0.044040sin(5𝑡)
	+0.002384sin(7𝑡)	−0.000104sin(9𝑡)	+0.000006sin(11𝑡)
	−0.000000sin(13𝑡)	−0.000000sin(15𝑡)	+0.000000sin(17𝑡)
	−68.359632𝑡cos(𝑡)	+2.878461𝑡cos(3𝑡)	−0.079121𝑡cos(5𝑡)
	−0.008828𝑡cos(7𝑡)	+0.000335𝑡cos(9𝑡)	−0.000022𝑡cos(11𝑡)
	+0.000000𝑡cos(13𝑡)	+0.000000𝑡cos(15𝑡)	−0.000000𝑡cos(17𝑡)
	−64.047243𝑡sin(𝑡)	−1.871954𝑡sin(3𝑡)	−0.188447𝑡sin(5𝑡)
	+0.009868𝑡sin(7𝑡)	−0.000169𝑡sin(9𝑡)	−0.000004𝑡sin(11𝑡)
	+0.000000𝑡sin(13𝑡)	−0.000000𝑡sin(15𝑡)	−0.000000𝑡sin(17𝑡)
	+61.716104𝑡^2cos(𝑡)	−0.768222𝑡^2cos(3𝑡)	+0.295936𝑡^2cos(5𝑡)
	−0.011360𝑡^2cos(7𝑡)	+0.000383𝑡^2cos(9𝑡)	+0.000014𝑡^2cos(11𝑡)
	−0.000000𝑡^2cos(13𝑡)	+0.000000𝑡^2cos(15𝑡)	+69.633323𝑡^2sin(𝑡)
	+2.498056𝑡^2sin(3𝑡)	−0.113525𝑡^2sin(5𝑡)	−0.012762𝑡^2sin(7𝑡)
	+0.000674𝑡^2sin(9𝑡)	−0.000026𝑡^2sin(11𝑡)	−0.000000𝑡^2sin(13𝑡)
	+0.000000𝑡^2sin(15𝑡)	−40.085044𝑡^3cos(𝑡)	−0.028980𝑡^3cos(3𝑡)
	+0.047373𝑡^3cos(5𝑡)	+0.009129𝑡^3cos(7𝑡)	−0.000811𝑡^3cos(9𝑡)
	+0.000013𝑡^3cos(11𝑡)	+0.000000𝑡^3cos(13𝑡)	−46.345112𝑡^3sin(𝑡)
	+0.014214𝑡^3sin(3𝑡)	+0.328854𝑡^3sin(5𝑡)	−0.010719𝑡^3sin(7𝑡)
	+0.000291𝑡^3sin(9𝑡)	+0.000013𝑡^3sin(11𝑡)	−0.000000𝑡^3sin(13𝑡)
	+18.758768𝑡^4cos(𝑡)	−1.045383𝑡^4cos(3𝑡)	−0.199353𝑡^4cos(5𝑡)
	+0.009593𝑡^4cos(7𝑡)	−0.000040𝑡^4cos(9𝑡)	−0.000004𝑡^4cos(11𝑡)
	+21.493321𝑡^4sin(𝑡)	−0.827863𝑡^4sin(3𝑡)	−0.028074𝑡^4sin(5𝑡)
	+0.005286𝑡^4sin(7𝑡)	−0.000478𝑡^4sin(9𝑡)	+0.000002𝑡^4sin(11𝑡)
	−6.218301𝑡^5cos(𝑡)	+0.995799𝑡^5cos(3𝑡)	+0.007797𝑡^5cos(5𝑡)
	−0.003375𝑡^5cos(7𝑡)	+0.000105𝑡^5cos(9𝑡)	−7.652998𝑡^5sin(𝑡)
	+0.017785𝑡^5sin(3𝑡)	−0.070424𝑡^5sin(5𝑡)	+0.004954𝑡^5sin(7𝑡)
	+0.000022𝑡^5sin(9𝑡)	+1.549574𝑡^6cos(𝑡)	−0.229644𝑡^6cos(3𝑡)
	+0.028038𝑡^6cos(5𝑡)	−0.000898𝑡^6cos(7𝑡)	+1.852341𝑡^6sin(𝑡)
	+0.213697𝑡^6sin(3𝑡)	−0.008050𝑡^6sin(5𝑡)	−0.001065𝑡^6sin(7𝑡)
	−0.230798𝑡^7cos(𝑡)	−0.028635𝑡^7cos(3𝑡)	+0.000743𝑡^7cos(5𝑡)
	−0.309074𝑡^7sin(𝑡)	−0.025324𝑡^7sin(3𝑡)	+0.008028𝑡^7sin(5𝑡)
	+0.019906𝑡^8cos(𝑡)	+0.004745𝑡^8cos(3𝑡)	+0.017987𝑡^8sin(𝑡)
	−0.012705𝑡^8sin(3𝑡)	−0.000299𝑡^9cos(𝑡)	+0.001329𝑡^9sin(𝑡)