= Solution
Define the one-step <local truncation error> as the defect obtained by inserting the exact solution into an update:
$$
\delta_{n+1}=y(x_n+h)-y(x_n)-h f(x_n,y(x_n)).
$$
For a smooth solution, <Taylor's theorem> and $y'=f(x,y)$ give
$$
\boxed{\delta_{n+1}=\frac{h^2}{2}y''(x_n)+O(h^3)=O(h^2).}
$$
Thus \b[$m=2$ for the one-step defect convention]. The leading derivative can also be written $y''=f_x+f_yf$ by the chain rule.
Some conventions instead call $\tau_{n+1}=\delta_{n+1}/h$ the truncation error. That <normalized local truncation error> is $O(h)$, so \b[$m=1$ under the normalized convention]. Both describe the same <Euler one-step defect>; neither should be confused with an unqualified global error claim.
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