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The Emden-Fowler transformation turns radial scaling into translation in a logarithmic variable. For a radial Generalized Lane-Emden equationput $s=\log r$, $a=2/(p-1)$ and $u(r)=r^{-a}v(s)$. Direct differentiation givesThe primes in the second equation are derivatives with respect to $s$. The choice $ap=a+2$ makes every term carry the same power of $r$, leaving an autonomous ordinary differential equation.

A constant nonzero equilibrium $v$ becomes a singular power law $u(r)=C{r}^{-a}$, explaining the Singular homogeneous solution of the Lane-Emden equation. Phase-plane trajectories instead describe profiles whose behavior changes with radius. The transformation is useful because dilation of $r$ becomes a shift of $s$, which an autonomous equation handles naturally.

Seek a homogeneous solution $u(r)=c{r}^{-\beta }$. Matching the powers in $\Delta u+u^p=0$ gives $\beta p=\beta +2$, henceSincematching coefficients givesThus the positive Singular homogeneous solution of the Lane-Emden equation is