Symbolic Implementation of Preliminary Group Classiffication for Ordinary Differential Equations

Table 1

Invariant forms of 𝑓^𝑖 admitted by the equivalence transformations of (19) with 𝜆_𝑖 arbitrary constants.

Operator	Invariant form of 𝑓^𝑖	𝑓^2
𝐺1	𝜆𝑖𝑥^−(𝑖+3)	𝜆2𝑥^−5
𝐺4	Null
𝐺2+𝛼𝐺4	𝜆𝑖𝑥^(𝛼−(1/2))𝑖−(𝛼+(3/2))	𝜆2𝑥^𝛼−(5/2)
𝐺1+𝐺3+𝛽𝐺4	𝜆𝑖(𝑥−1)^(−(1+𝛽)𝑖+𝛽−3)/2(𝑥+1)^((𝛽−1)𝑖−𝛽−3)/2	𝜆2(𝑥−1)^−(𝛽+5)/2(𝑥+1)^(𝛽−5)/2
𝐺1+𝜀𝐺4	𝜆𝑖𝑥^−(𝑖+3)exp(𝜀(𝑖−1)/𝑥)	𝜆2𝑥^−5exp(𝜀/𝑥)