Euler's method setup How can we use Euler's method to approximate the solutions for the following IVP below:
$$y' = -y + ty^{1/2},\text{ with }1 \leq t \leq 2,\ y(1) = 2,$$
and with $h = 0.5$
The main concern is the organization, i.e., set up of it for this particular example.
And, if the actual solution to the IVP above is: 
$$y(t) = (t-2+\sqrt{2} \mathrm{e} \cdot \mathrm{e}^{-t/2})^2$$, then, how to compare the actual error and compare the error bound?
Thanks
 A: Fix a small step $h$ and iterate:
$$
t_0 = 1 \\
y_0 = y(1) = 2 \\
t_{n+1} =  t_{n} + h \\
y_{n+1} = y_{n} + h (-y_{n} + t_{n} y_{n}^{1/2})$$
and the value $y_{n} \approx y(t_n).$
Try different small step $h = 0.1, 0.01, \ldots,$ and compare for all values of $t_0, t_0 + h, t_0 + 2h, \ldots$ the accuracy between the actual solution $y(t_n)$ you have and $y_n$ you computed.
A: The exact solution to the stated IVP actually reads:
$$
     y(t) = \mathrm{e}^{1-t} \left( 1 + \sqrt{2} + \mathrm{e}^{(t-1)/2} \cdot(t-2) \right)^2
$$
Here is the implementation of the Euler's scheme in Mathematica, for $h=0.2$:
In[68]:= sol[t_] := E^(1 - t) (1 + Sqrt[2] + E^((t - 1)/2) (-2 + t))^2;

In[69]:= f[t_, y_] := t Sqrt[y] - y;

In[70]:= approx = 
 With[{h = 0.2}, 
  NestWhileList[# + {h, h Apply[f, #]} &, {1.0, 2.0}, 
   First[#] < 2.0 &]]

Out[70]= {{1., 2.}, {1.2, 1.88284}, {1.4, 1.83559}, {1.6, 
  1.84783}, {1.8, 1.91326}, {2., 2.02856}}

In[71]:= Table[{t, sol[t]}, {t, approx[[All, 1]]}] - approx

Out[71]= {{0., 4.44089*10^-16}, {0., 0.0339166}, {0., 0.0594082}, {0.,
   0.080083}, {0., 0.0983063}, {0., 0.115599}}

