> help euler_step

  the program takes the intial data y_0 at the initial time x_0 and 
  computes up to time x_final using n timesteps.  It creates
  two vectors, x and y, which you can then plot against each other
  plot(x,y)
 
  [x,y] = euler_step(y_0,x_0,x_final,n)

compute the approximate solution y1, and the exact solution y1_true.  We're
solving the problem y'=3y with initial data y(0)=4.
> [x1,y1] = euler_step(4,0,1,10);
> y1_true = 4*exp(3*x1);
> plot(x1,y1_true); hold on; plot(x1,y1,'o');
> [x2,y2] = euler_step(4,0,1,20);
> y2_true = 4*exp(3*x2);
> plot(x2,y2_true); hold on; plot(x2,y2,'ro');
> [x3,y3] = euler_step(4,0,1,40);
> y3_true = 4*exp(3*x3);
> plot(x3,y3_true); hold on; plot(x3,y3,'go');

Looking at the plots of the approximate solutions, they appear to be going
to the true solution.  Note that the various x and y vectors are of different
lengths...
> whos x* y*
  Name          Size         Bytes  Class

  x1            1x11            88  double array
  x2            1x21           168  double array
  x3            1x41           328  double array
  y1            1x11            88  double array
  y1_true       1x11            88  double array
  y2            1x21           168  double array
  y2_true       1x21           168  double array
  y3            1x41           328  double array
  y3_true       1x41           328  double array

Grand total is 219 elements using 1752 bytes

To reach time 1, we have to look at the 11th, 21st, and 41st components:
> x1(11)
ans =
     1
> x2(21)
ans =
     1
> x3(41)
ans =
     1

The errors are initially huge, but appear to be decreasing...
> y1(11) - y1_true(11)
ans =
  -25.1988
> y2(21) - y2_true(21)
ans =
  -14.8760
> y3(41) - y3_true(41)
ans =
   -8.1652

We check the ratio of the errors... they appear to be going to 2...
> (y1(11) - y1_true(11))/(y2(21)-y2_true(21))
ans =
    1.6939
> (y2(21)-y2_true(21))/(y3(41)-y3_true(41))
ans =
    1.8219

Compute more solutions with finer time-steps to see if it's really 
converging like O(dt).
> [x4,y4] = euler_step(4,0,1,80);
> y4_true = 4*exp(3*x4);
> [x5,y5] = euler_step(4,0,1,160);
> y5_true = 4*exp(3*x5);
> [x6,y6] = euler_step(4,0,1,320);
> y6_true = 4*exp(3*x6);

The ratios are all getting closer to 2.
> (y3(41)-y3_true(41))/(y4(81)-y4_true(81))
ans =
    1.9031
> (y4(81)-y4_true(81))/(y5(161)-y5_true(161))
ans =
    1.9493
> (y5(161)-y5_true(161))/(y6(321)-y6_true(321))
ans =
    1.9741