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

Table 2

Table of 𝜆-symmetry classification with Lie symmetry of path equation.

Function	Lie symmetries	Integration factor	𝜆-Symmetries
Arbitrary 𝑓(𝑦)	𝜉=𝑎,𝜂=0	𝑦𝜇=^′1+𝑦^′2	𝜆=1+𝑦^′2𝑓^′(𝑦)𝑓𝑦𝑦^′
𝑓(𝑦)=𝑘	𝜉=𝑥𝑦,𝜂=𝑦^2	1𝜇=𝑤^2𝑦	𝑦𝜆=^′𝑦
𝑓(𝑦)=𝑦^𝑛	𝜉=1,𝜂=0	𝑦𝜇=^′1+𝑦^′2	𝜆=𝑛(1+𝑦^′2)𝑦𝑦^′
1𝑓(𝑦)=𝑚𝑦+𝑛	𝜉=𝑚𝑥^32+𝑥(𝑛𝑦+𝑚𝑦^22)
	𝑚𝜂=−^2𝑥^44(𝑚𝑦+𝑛); +𝑛^2𝑦^2+𝑚𝑛𝑦^3+𝑚^2𝑦^4/4𝑚𝑦+𝑛	𝜇=𝑚𝑦+𝑛𝑤^2(𝑚𝑥^2+2𝑛𝑦+𝑚𝑦^2)	𝜆=2𝑛^2𝑦^′+2𝑚𝑛(𝑥+𝑦𝑦^′)(𝑚𝑦+𝑛)(2𝑛𝑦+𝑚(𝑥^2+𝑦^2)); +𝑚^2(2𝑥𝑦−𝑥^2𝑦1+𝑦^2𝑦^′)(𝑚𝑦+𝑛)(2𝑛𝑦+𝑚(𝑥^2+𝑦^2))
𝑓(𝑦)=𝑘Exp(𝛼𝑦)	𝜉=0,𝜂=𝑒^𝑦𝛼cos𝑥𝛼	𝜇=𝑒^−𝛼𝑦cos𝑥𝛼	𝜆=𝛼(𝑦^′−tan𝑥𝛼)