How can i adapt Euloer backward method

Solve this by eulor backward method

then I write BW code in matlab simply

% dy/dt=y-t^2+1 ; 0<=t<=2 ; y(0)=0.5;
clc,clear;close all;
 f = @(t,y) (y-t^2+1); 
 a = 0; %input('initial ponit, a:  ');
 b = 100; %input('end point, b:  ');
 n = 10; %input('intervals, n: ');                
 alpha = 166*10^(4); %input('initial condition, alpha:  ');
  h = (b-a)/n;                                                
 t=[a zeros(1,n)]; 
 y=[alpha zeros(1,n)];
 for i = 1:n+1 
 t(i+1)=t(i)+h;
 yprime=y(i)+h*f(t(i),y(i)); %slope
 y(i+1)=y(i)+h*f(t(i+1),yprime);
 delta=y(i+1)-yprime;
 fprintf('%.4f  %.4f\n', t(i), y(i));
 figure(1)
 plot(t(i),y(i),'r*'); 
 grid on;
 plot(t(i),delta,'g*',t(i),yprime,'b*');
 xlabel('t values'); ylabel('y values');
 hold on;
 legend('delta','yprime','BW')
 end

0.0000  1660000.0000

Warning: Ignoring extra legend entries.

10.0000  184259110.0000
20.0000  20452747320.0000
30.0000  2270254903630.0000
40.0000  251998294197040.0000
50.0000  27971810655686552.0000
60.0000  3104870982780921344.0000
70.0000  344640679088681844736.0000
80.0000  38255115378843688370176.0000
90.0000  4246317807051649023213568.0000
100.0000  471341276582733031376158720.0000

then how can i adapt it???

[Torsten]

The code you use isn't Euler backward.

And the formula to compute N''_exist,i is already given (you wrote it in your question). So there is no need to use an Euler backwards integrator from the File Exchange (or whereever you got the code from).