= Emden-Fowler transformation
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The <Emden-Fowler transformation> turns radial scaling into translation in a logarithmic variable. For a radial <Generalized Lane-Emden equation>
$$
u''+\frac{d-1}{r}u'+u^p=0,\qquad r>0,\quad p>1,
$$
put $s=\log r$, $a=2/(p-1)$ and $u(r)=r^{-a}v(s)$. Direct <differentiation> gives
$$
v''+(d-2-2a)v'-a(d-2-a)v+v^p=0.
$$
The 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 $u(r)=Cr^{-a}$, 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.
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