= Solution
Regularity at the centre requires finite $\rho_c,P_c$ and $M(0)=0$. Then $M(r)=4\pi\rho_c r^3/3+o(r^3)$, so $P'(0)=0$. At a free stellar surface, $P(R)=0$; more generally it matches the external <pressure>. For $P=K\rho^2$ with $K>0$ and vacuum outside, this also gives $\rho(R)=0$.
Inside the star, cancel positive $\rho$ to obtain $2K\rho'=-GM/r^2$. Differentiate and use <mass conservation>:
$$
\frac1{r^2}\frac d{dr}(r^2\rho')=-\frac{2\pi G}{K}\rho.
$$
Thus the <Lane-Emden equation> of <polytropic index> one is
$$
\boxed{\frac1{x^2}\frac d{dx}\left(x^2\frac{d\rho}{dx}\right)=-\rho,\qquad x=\alpha r,\quad\alpha=\sqrt{\frac{2\pi G}{K}}.}
$$
For example, the regular solution is $\rho=\rho_c\sin x/x$, with its central value interpreted by <continuity>, and the first vacuum surface is at $x=\pi$. This also verifies the stated central and surface <boundary conditions>.
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