Xini[t_] := Sin[t]
Yini[t_] := Cos[t]

a := 5
b := 0

{sol, graf}=NDSolve[{x'[
                t] == ((Xini[t] - 
                        x[t]) √((((Xini'[t]))^2) + (((Yini'[t]))^2)))/(√((((Xini[t] - x[t]))^2) + (((Yini[t] - y[t]))^2))), y'[
                t] == ((Yini[t] - 
                        y[t]) √((((Xini'[t]))^2) + (((Yini'[t]))^2)))/(√((((Xini[t] - x[t]))^2) + (((Yini[t] - y[t]))^2))), {x[0] == a, y[0] == b}},{x[t], y[t]},{t, 0, 13},PlotSolution -> True,HowManyArrows -> 4,HeadLength -> 1/50,AspectRatio -> Automatic,Ticks -> {{-1, 1}, {-1, 1}},PlotRange -> Automatic]

{xsol[t_], ysol[t_]} = {x[t], y[t]} /. sol[[1]]

fig[
        t_] := 
Draw2D[{ParametricDraw[{xsol[s], 
              ysol[s]}, {s, 0, t}], 
          ParametricDraw[{Xini[s], Yini[s]}, {s, 0, t}], {PointSize[ .02], 
            Point[{xsol[t], ysol[t]}], PointSize[ .02], 
            Point[{Xini[t], 
                Yini[t]}]}, 
{Dashing[{ .01,  .01}], 
            Line[{{xsol[t], ysol[t]}, {Xini[t], 
                  Yini[t]}}]}, 
Thickness[ .001], 
Arrow[{xsol[t], ysol[t]}, {xsol[t], ysol[t]} + 
              1/2 {xsol'[t], ysol'[t]}], Thickness[ .001], 
          Arrow[{Xini[t], Yini[t]}, {Xini[t], Yini[t]} + 
              1/2 {Xini'[t], Yini'[t]}]}, 
        AspectRatio -> Automatic, 
        PlotRange -> {{Min[a, -1] -  .5, 
              Max[a, 1] +  .5}, {Min[b, -1] -  .5, 
              Max[b, 1] +  .5}}]

fig[1];

Do[fig[t], {t, 0.1, 12,  .1}];