First you take that second-order ODE apart into two first-order ODEs, 
as in one of the previous homework problems:

1. get highest-order term by itself on LHS:

  x'' = t - xx'

2. define helper var:  

  v=x'

3. rewrite and stick on helper var eqn:

  v' = t - xv
  x' = v

The state vector has two variables in it: [x]
                                          [v]

At t=0, this vector has the following entries:

[x]    = [1]
[v]      [2]
   t=0

At t=0, the derivative vector is:

[x']    = [   2    ]
[v']      [0 - 2(1)]
   t=0

Here's the forward euler algorithm, first step:

   [x]    = [1] + 0.1 [ 2] = [1.2]
   [v]      [2]       [-2]   [1.8]
     t=0.1

Next step:

   [x]    = [1.2] + 0.1 [     1.8      ] = [ 1.38]
   [v]      [1.8]       [0.1 - 1.8(1.2)]   [1.594]
     t=0.2