(* Symbolic solution *)
DSolve[{y'[t] == t*Sqrt[y[t]], y[0] == 1}, y, t]
Table[{t, 1/16 (4 + t^2)^2}, {t, 0, 10}]

(* Numerical solution I (not RK4) *)
Table[{t, y[t], Abs[y[t] - 1/16*(4 + t^2)^2]}, {t, 0, 10}] /.
 First@NDSolve[{y'[t] == t*Sqrt[y[t]], y[0] == 1}, y, {t, 0, 10}]

(* Numerical solution II (RK4) *)
f[{t_, y_}] := {1, t Sqrt[y]}
h = 0.1;
phi[y_] := Module[{k1, k2, k3, k4},
  k1 = h*f[y];
  k2 = h*f[y + 1/2 k1];
  k3 = h*f[y + 1/2 k2];
  k4 = h*f[y + k3];
  y + k1/6 + k2/3 + k3/3 + k4/6]
solution = NestList[phi, {0, 1}, 101];
Table[{y[[1]], y[[2]], Abs[y[[2]] - 1/16 (y[[1]]^2 + 4)^2]},
  {y,  solution[[1 ;; 101 ;; 10]]}]