Past exam of the topics in applied mathematics course of the University of Oxford / 2007 / A / 4 / iii / Solution

The cosine modes satisfy both zero-flux endpoint conditions and ${\partial }_x^2\cos (k_{n}x)=-k_n^2\cos (k_{n}x)$. 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 yieldsThe 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.