cdf =Manipulate[Column[{ SuFunc[ff, aa, yy0, nn]}],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,4,"Orden n="},ControlType->InputField},
Button["OK", ff=f;aa=a;yy0=y0; 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`*"];
escriba[lista_]:=Style[Row@lista,Black];
(*($PreRead=(#/. s_String/;StringMatchQ[s,NumberString]&&Precision@ToExpression@s==MachinePrecision:>s<>"`12."&));*)
ff=f;aa=a;yy0=y0; nn=n;
proposito ={"Propósito: Hallar un polinomio de Taylor de orden n que\naproxime la solución de la EDO y'=f(t, y)",  ", y(a)=", (Subscript[y, 0]),  ", con ", "a≤ t≤ b"};
SuFunc[f_, a_, y0_, n_] := Module[{r, coef, aux1, aux2, aux, sust},  

g[x_] := f/.{t-> x, y-> y[x]};
r ={"y(", a, ") = ", y0, "\n", y',"(", a,")"," = ", g[a]/.{y[a]->y0}, "\n"};
coef ={y0, g[a]};  
sust ={t->a, y[a]->y0, y'[a]->g[a]};
Do[
aux1[t_] = D[g[t], {t, i }];
aux =aux1[a]//.sust;
aux2[t_] = D[y[t], {t, i+1}]; 
sust =Join[sust, {aux2[a] ->aux1[a]}];
coef = Join[coef, {aux}]/.sust;    r=Join[r, {TraditionalForm[D[y[x], {x, i+1}]]/.x->a," = ",TraditionalForm@(Expand@aux1[t]/.t->a),"=",aux, "\n"}], {i, 1, n-1}]; r= Join[r,{"P(t) =",  (∑_(i=0)^n)((coef[[i+1]] (((t-a))^i))/((i)!))}];
escriba@r
];

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