Euler' s Method

g[y_,t_]=3+t-y

tzero=0
yzero=1

Here is the Euler method applied with stepsize 0.1

h=0.1

For[j=1;soln={{tzero,yzero}},j<5,j++,soln=Union[{{tzero+j h,soln[[j]][[2]]+g[soln[[j]][[2]],soln[[j]][[1]]] h}},soln]]

approx1=soln

Here is a way to figure out how arrays work :

testarray={{1,2},{3,4},{5,6}}

testarray[[1]]
testarray[[2]][[2]]

Here is the Euler method applied with stepsize 0.05

h=0.05

For[j=1;soln={{tzero,yzero}},j<9,j++,soln=Union[{{tzero+j h,soln[[j]][[2]]+g[soln[[j]][[2]],soln[[j]][[1]]] h}},soln]]

approx2=soln

Here is the Euler method applied with stepsize 0.025

h=0.025

For[j=1;soln={{tzero,yzero}},j<17,j++,soln=Union[{{tzero+j h,soln[[j]][[2]]+g[soln[[j]][[2]],soln[[j]][[1]]] h}},soln]]

approx3=soln

Show[ListPlot[approx1],ListPlot[approx2],ListPlot[approx3]]

The "actual" solution

nsoln=NDSolve[{ny'[t]==3+t-ny[t],ny[0]==1},ny,{t,0,0.4}]

Show[ListPlot[approx1],Plot[ny[t]/.nsoln,{t,0,0.4}]]