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If you change {x[t], z[t]} to {x,t} in NDSolve, then you'll get better results.  However, your functions are 0 everywhere, which you can see by setting the AxesOrigin of the plot to {0,-1} [mcode]c0 = 1; \[Omega] = 1; vg = 1; vp = 1; \[Sigma] = 1;...