= Solution
In <synchronous gauge in cosmology>, $h_{0\mu}=0$, so the explicit metric terms vanish. Symmetry of the covariant <energy-momentum tensor> and the background metric imply
$$
\delta T^j{}_0=-a^{-2}\delta^{ji}\delta T^0{}_i.
$$
Hence the two expansion terms in part (c) combine to $3H\delta T^0{}_j$. Substituting the <scalar velocity potential> and <scalar anisotropic stress> definitions gives
$$
\boxed{\partial_0[(\bar\rho+\bar P)\partial_j\delta u]
+3H(\bar\rho+\bar P)\partial_j\delta u
+\partial_j\delta P
+\partial_j\nabla^2\pi^S=0.}
$$
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