Troubles with the results of NDSolve

Question

Trying to build a parametric plot based on a NDSolve results, the Mathematica gives an error:

ReplaceAll::reps: {sol} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing

Can someone tell me what's the matter and how can I fix it?

Clear["Global`*"]
m = 10^-30; 
n = 8;
l = 650*10^-9;
k = 2 Pi/l;
a = 10^-5;
A = 0.01;
T = 10;
vo = (k/a)10^-2;
b = (10^-4)(k/a)^2/2n;

eq = x''[t]-b(1 + t)(Sin[x[t]] - A/k);

sol = 
  NDSolve[{eq == 0, x'[0] == vo, x[0] == 0}, x[t], {t, 0, T}, 
    MaxSteps -> 10^10, AccuracyGoal -> 10, MaxStepSize -> 0.001];

ParametricPlot[
  Evaluate[{x[t], x'[t]} /. sol], {t, 0, T}, 
    AxesLabel -> {"x(t)", "v(t)"}, 
    PlotRange -> {Full, Full}, 
    AspectRatio -> 1/2]

Answer 1

Looking at the code, you are solving for x[t], but the replacement you use requires you to solve for x (or vice versa). Replace x[t] with x in your `NDSolve' and your replacement will work.

sol = NDSolve[{eq == 0, x'[0] == vo, x[0] == 0}, x, {t, 0, T}, MaxSteps -> 1000, AccuracyGoal -> 10, MaxStepSize -> 0.001];