= Solution
Apply the $a=0$ method to the <Dahlquist test equation> and write $z=h\lambda$. The recurrence becomes
$$
\left(1-\frac32z^2\right)y_{n+2}-y_{n+1}-zy_n=0,
$$
so every amplification root must satisfy
$$
\left(1-\frac32z^2\right)\xi^2-\xi-z=0.
$$
Choose $z=-1$, which lies strictly in the left half-plane. The polynomial then gives $\xi^2+2\xi-2=0$, hence
$$
\xi=-1\pm\sqrt3.
$$
One root has modulus $1+\sqrt3>1$. Generic initial perturbations excite that growing recurrence mode even though the exact scalar solution decays. \b[The $a=0$ method is not A-stable.] There is also a singular implicit coefficient at $z=-\sqrt{2/3}$, another obstruction inside the left half-plane.
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