x0=0.1;y0=0.1;tend=2.0;

sol=NDSolve[{
x'[t]==Cos[t],
y'[t]==Cos[t]+Exp[t]*(y[t]-Sin[t]),
x[0]==x0,y[0]==y0},{x,y},{t,0,tend}]

Plot[Evaluate[{x[t],y[t]}/.sol],{t,0,tend}]

Clear[x1];Clear[x2];x1[0]=x0; x2[0]=y0; nn=1000;tau=tend/nn;

For[n=0, n<nn, n++,
{
tn1=(n+1)*tau;
sol=Solve[
{
x1[n+1]==x1[n]+tau*(Cos[tn1]),
x2[n+1]==x2[n]+tau*(Cos[tn1]+Exp[1.0*tn1]*(x2[n+1]-Sin[tn1]))
},
{x1[n+1],x2[n+1]}
];
x1[n+1]=x1[n+1]/.sol[[1,1]];
x2[n+1]=x2[n+1]/.sol[[1,2]];
}]

ListLinePlot[Table[{n*tau,x2[n]},{n,0,nn}]]

ListLinePlot[Table[{n*tau,x1[n]},{n,0,nn}]]