cdf =Manipulate[Column[{ SuFunc[ff, aa, yy0, nn, mm]}],Style[escriba@proposito,Bold,10],Spacer[{100,3}],
 (*Inicio contadores *)
 {{f,5t+3y,"f = "},ControlType->InputField},
{{a,2,"a = "},ControlType->InputField},
{{y0,1,"y(a) = "},ControlType->InputField},
{{b,3,"b = "},ControlType->InputField},
{{n,2,"n = "},ControlType->InputField},
{{m,4,"Orden m = "},ControlType->InputField},
Button["OK", ff=f; aa=a; bb=b; yy0=y0; mm=m; nn=n; ]
(* Fin contadores *)
,Alignment->Center,Deployed->True,AppearanceElements->"ResetButton",LocalizeVariables->True,ContinuousAction->False,FrameLabel->{{None,None},{Column[{,Row[{Style["Autores: Juan Félix Ávila Herrera y Enrique Vílchez Quesada",Bold,10,RGBColor[0,0,0.5]]}],Row[{Style["Escuela de Informática | Universidad Nacional de Costa Rica",Bold,10,RGBColor[0.5,0.1,0]]}]},Center],Style["Título: Método de Taylor para EDO.", Bold,14, RGBColor[0,0.33,0.33]]}},
Initialization:>(
ClearAll["Global`*"];
ff=f; aa=a; bb=b; yy0=y0; mm=m; nn=n;
siga =True;
escriba[lista_]:=Style[Row@lista,Black];
($PreRead=(#/. s_String/;StringMatchQ[s,NumberString]&&Precision@ToExpression@s==MachinePrecision:>s<>"`12."&));
proposito ={"Propósito: Usar el método de Taylor de orden m\npara aproximar la solución de la EDO y'=f(t, y)",  ", y(a)=", (Subscript[y, 0]),  ",\ncon ", "a≤ t≤ b. Se usa h =((b - a)/n)"};
SuFunc[f_, a_, y0_, n_, m_] := Module[{r={},  ww=a, h =((b-a)/n), pts={}, pol, coef, der}, 
g[x_] := f/.{t->x, y-> y[x]};
aux=g[t];
der =Table[aux =Expand[D[g[t], {t, i}]/.{D[y[t], {t, i}]->aux}],{i, 1, m-1}];
auxi = der;

r ={"h = ", N@h, "\n"};
ww= y0; 
der = Join[{g[t]}, der]; 
pts ={{a, y0}};
lw ={ww};
 
 
Do[
coef =der//.{ t->(a+ i *h), y[a+ i *h]->ww};

ww  = ww+(∑_(k=1)^m)(((h^k)/(k!)) coef[[k]])  +0.;
lw =AppendTo[lw, ww];
pts =Join[pts, {{a+(i+1)*h, ww}}];
 
, {i, 0, n-1}];
pol = Expand@ InterpolatingPolynomial[pts,t];
r=Join[r, Flatten@Table[{TraditionalForm[D[y[t], {t, i}]]," = ", der[[i]], "\n"}, {i, Length@der}]/.{y[t]->y}];
 r= Join[r,{Grid[Transpose@N@pts, Frame->All], "\n"}]; 
  r= Join[r,{"Polinomio interpolante:\n", "P(t) =", pol}]; 
(*If[siga,siga =False; escriba@r]*)
escriba@r
];

)]
(*CloudDeploy[cdf, Permissions->"Public"]*)