Linearization of third-order ordinary differential equations by generalized Sundman transformations: The case X‴+αX=0X‴+αX=0

Necessary and sufficient conditions which allow the most general third-order ordinary differential equation x‴=f(t,x,x′,x″)x‴=f(t,x,x′,x″) to be transformed to X‴+αX=0X‴+αX=0 under the generalized Sundman transformationX=F(t,x),dT=G(t,x)dt,(FxG≠0),are obtained in the paper. Here α≠0 is constant.