#
# Numerical Mathematics and Computing, Fifth Edition
# Ward Cheney & David Kincaid
# Brooks/Cole Publ. Co.
# (c) 2003
#
# file:  ode3
#
# Here is a Maple code which works out the initial value problem:
# x'(t) = 1 + x^2 + t^3, with initial condition x(1) = -4, using 
# the Taylor series method of order 4.

 y := array(1..4);
 Digits := 24;
 n := 100;
 h := 0.01;
 T := 1;
 X := -4;
# f is a function of x and t
 f := (x,t) -> 1 + (x(t))^2 + t^3;
# differentiate f with respect to t
 one := 1 + (x(t))^2 + t^3;
 two := diff(f(x,t),t);
 three := diff(%,t);
 four := diff(%,t);
 first := diff(x(t),t);
 second := diff(%,t);
 third := diff(%,t);
 for k from 1 to n do
   y[1] := subs(t = T, x(T) = X, one);
   y[2] := subs(first = y[1], t = T, x(T) = X, two);
   y[3] := subs({first = y[1], second = y[2]}, t =T, x(T) = X, three);
   y[4] := subs({first = y[1], second = y[2], third = y[3]}, t=T,x(T)=X,four);
   X := X + sum(y[i]*h^i/factorial(i), i=1..4);
   T := T + h;
 end do;