= Solution
If $D_1=D_2=D$, the linearized <matrix> for wavenumber $k$ is
$$
J-Dk^2I.
$$
Its <eigenvalues> are those of the homogeneous Jacobian shifted left by $Dk^2$, so a stable homogeneous equilibrium remains stable for every spatial mode. Thus equal diffusivities cannot produce a Turing instability. In part (b), equal diffusivities mean $\lambda=1$; because homogeneous stability requires $\beta>1$, the necessary condition $1-\lambda\beta>0$ fails, consistently ruling out instability.
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