= Solution
The <Dahlquist equivalence theorem> states that a linear multistep method is convergent for well-posed sufficiently smooth initial-value problems, with consistent starting values, if and only if it is consistent and zero-stable. Zero-stability means that every root of its first characteristic polynomial lies in the closed unit disk, with roots on the unit circle simple.
Here the characteristic polynomials are
$$
\rho(z)=z^3+(2a-3)z^2-(2a-3)z-1=(z-1)[z^2+2(a-1)z+1],\qquad\sigma(z)=a(z^2+z).
$$
Consistency holds because $\rho(1)=0$ and $\rho'(1)=\sigma(1)=2a$. The two remaining roots have product one. For $0<a<2$ they are distinct conjugate unit roots, and neither equals one. At $a=0$, the root one is triple; at $a=2$, the root minus one is double. Outside $[0,2]$, the remaining roots are real reciprocal roots and one has modulus greater than one. Thus
$$
\boxed{\text{convergence holds exactly for }0<a<2.}
$$
The order conditions give $c_2=0$ and $c_3=6-a$, which is nonzero throughout the convergent range. Therefore \b[the order of convergence is exactly two], for starting approximations of matching accuracy.
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