To make Euler's method more precise we can obviously decrease \( h \) (increase \( N \)). However, if we are computing the derivative \( f \) numerically by for example the two-steps formula $$ f'_{2c}(x)= \frac{f(x+h)-f(x)}{h}+O(h), $$ we can enter into roundoff error problems when we subtract two almost equal numbers \( f(x+h)-f(x)\approx 0 \). Euler's method is not recommended for precision calculation, although it is handy to use in order to get a first view on how a solution may look like. As an example, consider Newton's equation rewritten in Eqs. (8) and (9). We define \( y_0=y^{(1)}(t=0) \) an \( v_0=y^{(2)}(t=0) \). The first steps in Newton's equations are then $$ \begin{equation} y^{(1)}_1=y_0+hv_0+O(h^2) \tag{18} \end{equation} $$ and $$ \begin{equation} y^{(2)}_1=v_0-hy_0k/m+O(h^2). \tag{19} \end{equation} $$