= Euler one-step defect
{c}
{title2=$\delta_h=y(x+h)-y(x)-h f(x,y(x))=\tfrac12h^2y''(x)+O(h^3)$}
For a smooth solution of $y'=f(x,y)$, <Taylor's theorem> gives the displayed <local truncation error> when an <Euler method> step starts at an exact solution value. Thus the one-step defect is $O(h^2)$. Dividing it by $h$ gives a <normalized local truncation error> of $O(h)$; these are different conventions for reporting the same residual. Also $y''=f_x+f_yf$ along the solution by the <chain rule>. A global $O(h)$ error conclusion additionally requires the usual regularity and stability control over successive steps.
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