= Solution
Fix the following normalization of the <local truncation error>, also called the <consistency error>: insert a smooth exact solution into the <linear multistep method> and divide the residual by $h$,
$$
T_n(h)=\frac1h\sum_{j=0}^k\alpha_jy(x_n+jh)-\sum_{j=0}^k\beta_jy'(x_n+jh).
$$
Thus the unscaled defect is $hT_n(h)$. <Consistency of a numerical method> means that $T_n(h)\to0$ uniformly over mesh points in each fixed interval for every sufficiently smooth exact solution. Write the <characteristic polynomials of a linear multistep method> as
$$
\rho(z)=\sum_{j=0}^k\alpha_jz^j,\qquad \sigma(z)=\sum_{j=0}^k\beta_jz^j.
$$
By <Taylor's theorem>, <consistency of a numerical method> is equivalent to
$$
\sum_j\alpha_j=0,\qquad \sum_jj\alpha_j=\sum_j\beta_j,
$$
or $\rho(1)=0$ and $\rho'(1)=\sigma(1)$.
Second-order accuracy means $T_n(h)=O(h^2)$ for smooth solutions, equivalently an unscaled <local truncation error> $O(h^3)$. In addition to the two consistency conditions, its order conditions require
$$
\sum_jj^2\alpha_j=2\sum_jj\beta_j.
$$
An <order of a numerical method> of exactly two additionally requires that the next order condition fails. These statements concern the local defect; a second-order <global error> requires stability and suitably accurate starting values as well.
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