= Solution
For the constant-density model, $m(r)=4\pi\rho r^3/3=Mr^3/R^3$. This is the incompressible <Interior Schwarzschild metric>, replacing the earlier finite-slope <barotropic stellar equation of state>. The <Tolman–Oppenheimer–Volkoff equation> separates as
$$
\frac{dp}{(p+\rho)(p+\rho/3)}=-\frac{4\pi r\,dr}{1-8\pi\rho r^2/3}.
$$
On the left, partial fractions give
$$
\int\frac{dp}{(p+\rho)(p+\rho/3)}=\frac{3}{2\rho}\log\frac{p+\rho/3}{p+\rho}.
$$
On the right the antiderivative is $\tfrac3{4\rho}\log(1-8\pi\rho r^2/3)$. Applying the surface condition $p(R)=0$, and writing $s(r)=\sqrt{1-2Mr^2/R^3}$, $s_R=\sqrt{1-2M/R}$, gives
$$
\frac{p+\rho/3}{p+\rho}=\frac{s(r)}{3s_R}.
$$
Solving for the <pressure> recovers the <Interior Schwarzschild solution>:
$$
p(r)=\rho\frac{s(r)-s_R}{3s_R-s(r)}.
$$
In particular the <central pressure of a constant-density relativistic star> is
$$
\boxed{p_c=\rho\frac{1-\sqrt{1-2M/R}}{3\sqrt{1-2M/R}-1}.}
$$
The denominator must be positive for a regular positive-pressure centre, requiring $M/R<4/9$. As that limiting compactness is approached, the central <pressure> diverges, in agreement with <Buchdahl's theorem>.
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