= Solution
The cosine modes satisfy both zero-flux endpoint conditions and $\partial_x^2\cos(k_nx)=-k_n^2\cos(k_nx)$. Linearizing the reaction terms gives $f_a\delta a+f_b\delta b$ and $g_a\delta a+g_b\delta b$. Cancelling the common mode factor yields
$$
\boxed{\lambda\binom{a_n}{b_n}=\left[J-\begin{pmatrix}k_n^2&0\\0&Dk_n^2\end{pmatrix}\right]\binom{a_n}{b_n},\qquad k_n=n\pi/L.}
$$
The <derivatives> are exactly the entries of the matrix computed in (ii). The zero mode reproduces the stable homogeneous perturbation; positive integers give the possible spatial instabilities.
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