= Solution
The <Brans-Dicke theory> couples a <scalar field> $\Phi$ to the <Ricci scalar>. Assume $\Phi\ne0$, constant $\omega$, matter independent of $\Phi$, and variations of compact support so <boundary terms> may be discarded. Use the paper's <curvature sign convention> and signature $(+---)$. Set $X=g^{ab}\partial_a\Phi\,\partial_b\Phi$ and $\Box\Phi=\nabla_a\nabla^a\Phi$. Define the <stress-energy tensor> consistently by
$$
\delta S_m=\frac12\int\sqrt{-g}\,T_{ab}\,\delta g^{ab}\,d^4x
=-\frac12\int\sqrt{-g}\,T^{ab}\,\delta g_{ab}\,d^4x.
$$
This convention fixes the sign of the matter term.
Vary with respect to $g^{ab}$. The <variation of metric volume density> contributes $-g_{ab}/2$ times each <scalar> Lagrangian, while $\delta R=R_{ab}\delta g^{ab}+g^{ab}\delta R_{ab}$. Applying the <Palatini identity> followed by two rounds of <integration by parts> gives the <metric variation of a scalar-curvature coupling>,
$$
\delta\int\sqrt{-g}\,\Phi R\,d^4x
=\int\sqrt{-g}\left[
\Phi G_{ab}+\nabla_a\nabla_b\Phi-g_{ab}\Box\Phi
\right]\delta g^{ab}\,d^4x.
$$
The differentiated $\Phi$ terms remain because the <curvature> multiplier is not constant. Varying the kinetic term at fixed $\Phi$ gives
$$
\delta\int\sqrt{-g}\,\omega\Phi^{-1}X\,d^4x
=\int\sqrt{-g}\,\omega\Phi^{-1}
\left(\partial_a\Phi\,\partial_b\Phi-\frac12g_{ab}X\right)
\delta g^{ab}\,d^4x.
$$
Combining these terms with the matter variation, the <Euler-Lagrange equation> is
$$
\boxed{\Phi G_{ab}+\nabla_a\nabla_b\Phi-g_{ab}\Box\Phi
+\omega\Phi^{-1}
\left(\partial_a\Phi\,\partial_b\Phi-\frac12g_{ab}X\right)
=-8\pi G T_{ab}.}
$$
Raising both free indices reproduces the contravariant form, because $\nabla_ag_{bc}=0$.
The independent <scalar> variation gives
$$
R-\omega\Phi^{-2}X
-2\omega\nabla_a(\Phi^{-1}\nabla^a\Phi)=0,
$$
or $R+\omega\Phi^{-2}X-2\omega\Phi^{-1}\Box\Phi=0$. The <trace> of the <metric tensor> equation in four dimensions is
$$
-\Phi R-3\Box\Phi-\omega\Phi^{-1}X=-8\pi G T,
\qquad T=g^{ab}T_{ab}.
$$
Multiplying the <scalar> equation by $\Phi$ eliminates the same combination $\Phi R+\omega\Phi^{-1}X$. Thus the <scalar equation of Brans-Dicke theory> is
$$
\boxed{(3+2\omega)\Box\Phi=8\pi G T,\qquad
\Box\Phi=\frac{8\pi G}{3+2\omega}\,g_{ab}T^{ab}
\quad(\omega\ne-3/2).}
$$
At the <degenerate coupling of Brans-Dicke theory>, $\omega=-3/2$, the combined equations impose $T=0$ instead; division by $3+2\omega$ is unavailable. If matter depended directly on $\Phi$, its <scalar> variation would add a source and the displayed <scalar> equation would change.
The following lettered sections justify the supplied variation identities; they are supporting identities in the PDF, rather than three independent field-equation problems.
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