Free fall on a fixed center and harmonic oscillator.

Newton’s equation for the free fall (Unnecessary constants have been removed) :

DSolve[{x''[t]==-(1/((x[t])^2)),x[0]==1,x'[0]==0},x[t],t]

That equation is best integrated analytically by considering the energy first integral, (((x'[t])^2)/2-1/x=E (=-1 on account of the particular initial conditions choosen)). Here is the solution in implicit form t=time(x) :

time[x_]:=(( (Sqrt[x(1-x)])+  ArcCos[(Sqrt[x])])/((Sqrt[2]) ))

x[t_]:=InverseFunction[time][t]

D[x[t],{t,2}]==-(1/((x[t])^2))//FullSimplify

Plot[x[t],{t,0,(π/(2 (Sqrt[2])))},AspectRatio->Automatic,Axes->False,Frame->True,PlotStyle->Thick,FrameLabel->{"t","x(t)"},LabelStyle->Directive[Large,Bold]]

A forward numerical integration suspects the presence of a singularity near t=1.11... :

NDSolve[{x''[t]==-(1/((x[t])^2)),x[0]==1,x'[0]==0},x[t],{t,0,2}]

Changing both dependant and independant variables in Mathematica :

1) The change of Sundman (x=(X^2) and dt=(X^2) dT) :

((Dt[((Dt[((X[T])^2) ])/(((X[T])^2) Dt[T]))])/(((X[T])^2) Dt[T]))+(1/((X[T])^4))//Simplify

(1/2)(((((Dt[((X[T])^2) ])/(((X[T])^2) Dt[T]))))^2)-(1/((X[T])^2))==Energy//Simplify

Combining the two former equations leads to a harmonic oscillation of X as a function of T.

Plot[x[t],{t,0,(π/(2 (Sqrt[2])))},AspectRatio->Automatic,Axes->False,Frame->True,PlotStyle->Thick,FrameLabel->{"t","x(t)"},LabelStyle->Directive[Large,Bold]]

Plot[Piecewise[{{x[t],0<t<(π/(2 (Sqrt[2])))},{x[((2π)/(2 (Sqrt[2])))-t],(π/(2 (Sqrt[2])))<t <((2π)/(2 (Sqrt[2])))},{x[t-((2π)/(2 (Sqrt[2])))],((2π)/(2 (Sqrt[2])))<t<((3π)/(2 (Sqrt[2])))},{x[((4π)/(2 (Sqrt[2])))-t],((3π)/(2 (Sqrt[2])))<t <((4π)/(2 (Sqrt[2])))}}],{t,0,((4π)/(2 (Sqrt[2])))},AspectRatio->Automatic,Axes->False,Frame->True,PlotStyle->Thick,FrameLabel->{"t","x(t)"},LabelStyle->Directive[Large,Bold]]

2) No monomial point transformation, x=(X^a)(T^b) and t=(X^c)(T^d) (where a, b, c, d are constants whatever their numerical values) can achieve such a  reduction :

((Dt[((Dt[((X[T])^a) (T^b)])/(Dt[((X[T])^c) (T^d)]))])/(Dt[((X[T])^c) (T^d)]))+(1/(((X[T])^(2a)) (T^(2b))))/.{Dt[a]->0,Dt[b]->0,Dt[c]->0,Dt[d]->0}//Simplify

Together[(1/2)(((((Dt[((X[T])^a) (T^b)])/(Dt[((X[T])^c) (T^d)]))))^2)-(1/(((X[T])^a) (T^b)))/.{Dt[a]->0,Dt[b]->0,Dt[c]->0,Dt[d]->0}]//Simplify