Plot[Cos[t],{t,0,10π}]

$RecursionLimit =50000;
y[t_]:=y[t-1]-0.1 Sin[t-1]
y[0]=1;
ListLinePlot[Table[y[1,t],{t,0,10π}]]

(*$RecursionLimit =50000;*)
y[a_,t_]:=y[a,t-1]-a Sin[t-1]
y[a_,0]:=1;

y[.1,10]

Table[y[.1,i],{i,1,50}]

ListPlot[%6,Joined-> True]

ListPlot[%]

Manipulate[
ListLinePlot[Table[y[a,t],{t,0,10π}]],
{a,0.1,2,0.01}
]

Manipulate[ListLinePlot[{
Table[{t,Cos[t]},{t,0,10π,0.1}],Table[{t,y[a,t]},{t,0,10π}]
}],
{a,0.001,2,0.01}
]

ListLinePlot[
Table[{t,y[0.1,t]},{t,0,10π}]
]

RUNGE-KUTTA:

k1[a_,t_] := - a Sin[t]
k2[a_,t_] := - a Sin[t-(a/2)]
k3[a_,t_] := - a Sin[t-(a/2)]
k4[a_,t_] := - a Sin[t-a]
y[a_,t_] := y[a,t-1]