Page 111 Example 1

(* 
      the first integration required *) 
y2Int = Integrate[
        2 (1 + 
                (s/(s + 1))) - (((s + 
                    1))^2) ((((1/(s + 1)) + (s/(((s + 1))^2))))^2), {s, 
          0, x}, Assumptions -> {x  > 0}]

(* 
      an auxiliary function to performe the integrations in general *)

g[f_] := ((1/(x + 1)) +  (1/(((x + 1))^2)) 
              Integrate[
                2 (s + 1) f  - (((s + 1))^2) (f^2), {s, 0, x}, 
                Assumptions -> {x  >  0}]) /. {x -> 
            s}

g[
      (1/(1 + s))]

(* 
      compute many other iterates *) 
allIters = NestList[g, (1/(1 + s)), 5]

(* 
      plot each solution looking at the convergence *) 
Plot[
        Evaluate[allIters], {s, 0, 1}];