First Integrals, Integrating Factors, and Invariant Solutions of the Path Equation Based on Noether and 𝜆-Symmetries

Table 1

Noether symmetry classification table of path equation.

Function	Infinitesimals and first integrals
𝑓(𝑦)=𝑘	𝑐𝜉=^1+𝑐2𝑥+𝑐3𝑥^2𝑘^2𝑐,𝜂=^2𝑦+2𝑥𝑦𝑐3+2𝑘𝑐4+2𝑘𝑥𝑐52𝑘^2
	𝐼1=𝑐1(2ln𝑘−𝑦^′2)2𝑘^2, 𝐼2=𝑦𝑦^′−𝑥𝑦^′22𝑘^2,𝐼3=−𝑦^2+2𝑥𝑦𝑦^′−𝑥^2𝑦^′22𝑘^2
	𝐼4=𝑦^′𝑘,𝐼5=−2𝑘𝑦+2𝑘𝑥𝑦^′2𝑘^2
𝑓(𝑦)=𝑦	𝑐𝜉=^1𝑦^2,𝜂=0,𝐼1=−(1+𝑦^′2)2𝑦^2
𝑓(𝑦)=𝑘𝑒^𝛼𝑦	𝑒𝜉=^−2𝑦𝛼2𝑘^2𝛼(𝑐1sin2𝑥𝛼−𝑐2cos2𝑥𝛼+2𝑐3)
	𝑒𝜂=^−2𝑦𝛼2𝑘^2𝛼(2𝑐4𝑘𝛼𝑒^𝑦𝛼cos𝑥𝛼−𝑐1cos2𝑥𝛼+2𝑐5𝑘𝛼𝑒^𝑦𝛼sin𝑥𝛼−𝑐2sin2𝑥𝛼)
	𝐼1=𝑒^−2𝑦𝛼4𝑘^2𝛼(−2cos2𝑥𝛼𝑦^′−sin2𝛼𝑥(𝑦^′2−1))
	𝐼2=𝑒^−2𝑦𝛼4𝑘^2𝛼(−2sin2𝑥𝛼𝑦^′+cos2𝛼𝑥(𝑦^′2−1))
	𝐼3𝑒=−^−2𝑦𝛼2𝑘^2𝛼(1+𝑦^′2),𝐼4=𝑒^−𝑦𝛼𝑘(cos𝑥𝛼𝑦^′−sin𝑥𝛼),𝐼5=𝑒^−𝑦𝛼𝑘(cos𝑥𝛼+sin𝑥𝛼𝑦^′)
1𝑓(𝑦)=𝑚𝑦+𝑛	𝜉=(𝑚𝑦+𝑛)^2(𝑐1+𝑐2𝑥+𝑐3𝑥^2)
	3𝜂=(𝑚𝑦+𝑛)(−4𝑚𝑥^2𝑐2−12𝑚𝑥^3𝑐3+14𝑦(2𝑛+𝑚𝑦)(𝑐2+2𝑥𝑐3)+𝑐4+𝑥𝑐5)
	𝐼1=18(−8𝑚𝑛𝑦−4𝑚^2𝑦^2+8𝑛^21ln𝑚𝑦+𝑛+8𝑛^2ln(𝑚𝑦+𝑛)−4(𝑚𝑦+𝑛)^2𝑦^′2)
	𝐼2=18(−2𝑚^2𝑥^3+4𝑚𝑛𝑥𝑦+2𝑚^2𝑥𝑦^2−6𝑚𝑥^2𝑚𝑦+𝑛𝑦^′
	+2𝑚𝑦^2𝑚𝑦+𝑛𝑦^′−4𝑥(𝑚𝑦+𝑛)^2𝑦^′2)
	𝐼31=−8(−𝑚^2𝑥^4+4𝑚𝑛𝑥^2𝑦−4𝑛^2𝑦^2+2𝑚^2𝑥^2𝑦^2−4𝑚𝑛𝑦^3−𝑚^2𝑦^4−4𝑚𝑥^3𝑚𝑦+𝑛𝑦^′
	+8𝑛𝑥𝑦𝑚𝑦+𝑛𝑦^′+4𝑚𝑥𝑦^2𝑚𝑦+𝑛𝑦^′−4𝑥^2(𝑚𝑦+𝑛)^2𝑦^′2)
	𝐼4=𝑚𝑥+(𝑚𝑦+𝑛)𝑦,𝐼5=18(4𝑚𝑥^2−8𝑛𝑦−4𝑚𝑦^2+8𝑥𝑚𝑦+𝑛𝑦^′)
𝑓(𝑦)=𝑦^𝑛	𝜉=𝑐1𝑦^−2𝑛1,𝜂=0,𝐼=−2𝑦^−2𝑛(1+𝑦^′2)