function [t,w]=BackwardEulerfe(N,a,b,alpha,x)

%N=number of steps
%a=initial time
%b=final time
%alpha=initial values at t=0
%x=initial guesses of the function

%Forward Euler was initailly coded up to provide the initial guesses for
%the newton method. But was not later used.
% wfe=zeros(1,n);

% h=(b-a)/N;
% %w=zeros(1,:);
% w(1,:)=alpha;
% t(1)=a;
% n=length(alpha); %take in y_initial for all the different
% wfe=zeros(n,1);


%FORWARD EULER IS RAN 10 TIMES TO PROVIDE GUESSES FOR BACKWARD EULER
% h=(b-a)/10; 
% wfe(1,:)=alpha; %takes in y_intial
% T(1)=a;      %take in t_initial
% 
% 
% for i=1:10
%     wfe(i+1,:)=wfe(i,:)+h*(f_ForwardEuler(T(i),wfe(i,:)));
%     T(i+1)=T(i)+h;
% end;
% 
% x=wfe(10,:)



n=length(alpha); %take in y_initial for all the different
w=zeros(1,n);    %makes a zero function

h=(b-a)/N;            
w(1,:)=alpha;   %puts in the initial value
t(1)=a;         %initial value of the time 



for j=1:N
    
    w(j+1,:)=newton(x,w(j,:),h,t(j));  %Backward euler step
    t(j+1)=t(j)+h;
    
end;