Gronwall's Theorem

Needs["Graphics`Colors`"];

x0 = DSolve[{x'[t] == x[t] - Log[ t], x[1] == 1}, x[t], t]

x1 = DSolve[{x'[t] == x[t] - Log[ t], x[1] == 1.1}, x[t], t]

\(\(Plot[{Evaluate[x0[\([1, 1, 2]\)]], 
		Evaluate[x1[\([1, 1, 2]\)]], 
        
		Evaluate[x0[\([1, 1, 2]\)]] + .1  Exp[t], 
        Evaluate[x0[\([1, 1, 2]\)]] - .1  Exp[t]}, {t, 1, 5}, 
      PlotStyle -> {{AbsoluteThickness[2], Blue}, {AbsoluteThickness[2], 
            Green}, {AbsoluteThickness[2], Red}, {AbsoluteThickness[2], 
            Red}}]; \)\)