Group classification of ODE y‴=F(x,y,y′)y‴=F(x,y,y′)

•Group classification of third-order ODEs, not involving the second derivative, is carried out.•Lie algebra of the equivalence group for the equations in question is computed.•The results are compared with Lie’s similar results for second-order equations.

In his extensive work of 1884 on the group classification of ordinary differential equations Lie performed, inter alia, the group classification of the particular type of the second-order equations y″=F(x,y)y″=F(x,y). In the present paper we extend Lie’s classification to the third-order equations y‴=F(x,y,y′)y‴=F(x,y,y′).