= Solution
Write $\mathcal R=k/(\mu H)$, where $H$ is the atomic mass unit used in the <mean molecular weight> convention, and let $\kappa$ be the <Rosseland mean opacity> per unit mass. For a static spherical star with negligible thermal storage, the <stellar structure equations> are
$$
\boxed{\frac{dm}{dr}=4\pi r^2\rho,\qquad\frac{dP}{dr}=-\frac{Gm\rho}{r^2},\qquad\frac{dL}{dr}=4\pi r^2\rho\epsilon,\qquad\frac{dT}{dr}=-\frac{3\kappa\rho L}{16\pi ac\,r^2T^3},\qquad P=\mathcal R\rho T.}
$$
Here $\epsilon$ is the net <stellar energy-generation rate> per unit mass; $a$ is the radiation energy-density constant and $c$ the speed of light. The last differential equation follows from <radiative diffusion>, $F=-(c/3\kappa\rho)d(aT^4)/dr$, with $L=4\pi r^2F$. There is no gravothermal storage term in the specified <stellar energy balance equation>. Neglecting <radiation pressure> in force balance does not remove radiation as an energy carrier. Regular central conditions are $m(0)=L(0)=0$; take negligible external <pressure>, $P(R)=0$, for the analytic <hydrostatic equilibrium> model below.
Set $x=r/R$. Integrating the <enclosed mass> directly gives
$$
m(r)=4\pi\rho_c\left(\frac{r^3}{3}-\frac{r^4}{4R}\right),\qquad M=\frac{\pi\rho_cR^3}{3}.
$$
Thus
$$
\boxed{\rho_c=\frac{3M}{\pi R^3},\qquad m(r)=M(4x^3-3x^4).}
$$
The <stellar hydrostatic equation> then gives
$$
P(r)=4\pi G\rho_c^2R^2\int_x^1\left(\frac{t}{3}-\frac{t^2}{4}\right)(1-t)\,dt
=4\pi G\rho_c^2R^2\left(\frac5{144}-\frac{x^2}{6}+\frac{7x^3}{36}-\frac{x^4}{16}\right).
$$
Factorizing makes the surface behaviour transparent:
$$
\boxed{P(r)=P_c(1-x)^2\left(1+2x-\frac95x^2\right),\qquad P_c=\frac{5\pi G\rho_c^2R^2}{36}=\frac{5GM^2}{4\pi R^4}.}
$$
The <ideal gas> equation of state gives the <temperature> profile
$$
\boxed{T(r)=T_c(1-x)\left(1+2x-\frac95x^2\right),\qquad T_c=\frac{P_c}{\mathcal R\rho_c}=\frac{5GM}{12\mathcal R R}=\frac{5\mu HGM}{12kR}.}
$$
No small-$x$ approximation was needed. In particular, close to the centre,
$$
m=4Mx^3+O(x^4),\qquad\frac{P}{P_c}=1-\frac{24}{5}x^2+O(x^3),\qquad\frac{T}{T_c}=1+x-\frac{19}{5}x^2+O(x^3).
$$
These exact hydrostatic formulas reveal a limitation of the prescribed <linear-density stellar model>. Its <mass density> has a central cusp, and $T'(0^+)=T_c/R>0$. A positive opacity with positive outward <luminosity> would instead require $T'<0$ in <radiative diffusion>. Moreover, regular positive nuclear heating gives $L=O(r^3)$ and therefore $T'=O(r)$ at the centre. Thus the imposed profile is a \b[hydrostatic toy model], not a complete positive-heating radiative-equilibrium solution of all the <stellar structure equations>. The requested algebraic profiles and nuclear-luminosity integral can still be computed consistently as properties of that toy model.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-58-linear-density-profiles.png]
{title=Normalized mass, density, pressure and temperature in the linear-density hydrostatic stellar model}
{height=400}
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