= Solution
The <characteristic polynomials of a linear multistep method> are
$$
\rho(\zeta)=\zeta^2-1,\qquad \sigma(\zeta)=a\zeta^2+2(1-a)\zeta+a.
$$
The <consistency of a numerical method> conditions hold for every real $a$:
$$
\rho(1)=0,\qquad \rho'(1)=2=\sigma(1).
$$
Both <roots of a polynomial> of $\rho$, namely $1$ and $-1$, are simple and have <modulus> one. The <root condition for a multistep method> therefore gives <zero-stability> for every $a$. The <Dahlquist equivalence theorem> now gives \b[convergence for every fixed $a\in\mathbb R$]. As usual, this means <convergence of a numerical method> on each fixed finite interval for a sufficiently regular <ordinary differential equation> with a <Lipschitz continuous> <vector field> and starting values tending to the exact starting values. When $a\ne0$, the <implicit time-stepping method> update is locally uniquely solvable for sufficiently small $h$, for example by a <contraction mapping> if $|ha|L<1$. <Zero-stability> does not assert that a large fixed step is suitable for a <stiff differential equation>.
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