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

Table 3

𝜆-Symmetry classification table of path equation with Jacobi last multiplier.

Function	𝜆-Symmetries with Jacobi last multiplier
𝑓(𝑦)=𝑘	𝜉^(𝜆)=𝑦(𝑐2+𝑐1𝑥)+𝑐3𝑥^2+𝑐8𝑥+𝑐7
	𝜂^(𝜆)=𝑦(𝑐4+𝑐3𝑥)+𝑐5𝑥+𝑐1𝑦^2+𝑐6
𝑓(𝑦)=𝑦	𝜉^(𝜆)=𝑐1𝑦^2,𝜂^(𝜆)=0
	𝜉^(𝜆)=𝑒^−3𝑦𝛼(𝑐1cos𝑥𝛼+𝑐2𝑒sin𝑥𝛼+^𝑦𝛼𝛼(2𝑐4𝛼−𝑐5cos2𝑥𝛼+𝑐6sin2𝑥𝛼))
𝑓(𝑦)=𝑘𝑒^𝛼𝑦	𝜂^(𝜆)=𝑒^−3𝑦𝛼(𝑐1sin𝑥𝛼−𝑐2cos𝑥𝛼+𝑒^2𝑦𝛼(𝑐7cos𝑥𝛼+𝑐8sin𝑥𝛼)
	+𝑒^𝑦𝛼(𝑐3−1𝑐2𝛼^6cos2𝑥𝛼+𝑐5sin2𝑥𝛼))
1𝑓(𝑦)=𝑚𝑦+𝑛	𝜉^(𝜆)=(𝑚𝑦+𝑛)^2(𝑐1𝑚^2𝑥^3+3𝑐2𝑚^2𝑥^2+2𝑐3𝑥^2+𝑐6𝑥+𝑐5)+(𝑚𝑦+𝑛)^4(𝑐1𝑥+𝑐2)
	𝜂^(𝜆)=12(𝑚𝑦+𝑛)(2𝑐7+𝑐1(𝑚𝑦+𝑛)^2(𝑚^2𝑦^2+2𝑚𝑛𝑦−𝑛^2)𝑚+2(𝑚𝑦+𝑛)^2𝑎4+𝑎3𝑥 )
𝑓(𝑦)=𝑦^𝑛,𝑛>1	𝜉^(𝜆)=𝑐1𝑦^((𝑛−1)2√(3𝑛−1)/3𝑛−1)+(1/2)(1−5𝑛−((𝑛−1)2√(3𝑛−1)/3𝑛−1)),𝜂^(𝜆)=0
𝑓(𝑦)=𝑦^𝑛,𝑛=1/3	𝜉^(𝜆)=−3𝑐12𝑦^2/3,𝜂^(𝜆)=0
𝑓(𝑦)=𝑦^𝑛,𝑛=1/2	𝜉^(𝜆)=𝑐1𝑦,𝜂^(𝜆)=0