DSolve[{y''[x]==k*Sin[2*(y[x]+o*x)]},{y},x]

Plot3D[JacobiAmplitude[x,y],{x,0,10},{y,0,10}]

Simplify[-o x-JacobiAmplitude[x (Sqrt[-1+(o^2)]),-(2/(-1+(o^2)))]]

DSolve[{R''[x]==R[x]*Cos[2*(-o x-JacobiAmplitude[(Sqrt[-1+(o^2)]) x,-(2/(-1+(o^2)))]+o*x)]},{R},x]

ss=NDSolve[{y''[x]==Sin[2(y[x]-omega*x)],y'[0]==0,y[0]==0,R''[x]==R[x]*(Cos[y[x]-omega*x]^2-Sin[y[x]-omega*x]^2),
R'[0]==0,R[0]==0.1}/.omega->100,{R,y},{x,0,10}];
Plot[Evaluate[{R[x]}/.ss],{x,0,10},PlotRange->All,AxesLabel->{"t","R"}]

Export["graf.pdf",%]