= Power substitution for an oscillatory second-order equation
For $\gamma\ne0$ and $x>0$, the <change of variables> $u=x^\gamma$ converts
$$
x^{2-2\gamma}y''+(1-\gamma)x^{1-2\gamma}y'+\gamma^2y=0
$$
into $Y_{uu}+Y=0$, where $y(x)=Y(u)$. The <chain rule> cancels the first-derivative terms. Thus $\cos(x^\gamma)$ and $\sin(x^\gamma)$ form a pair of <linearly independent> solutions, with <Wronskian> $\gamma x^{\gamma-1}$. This is a variable-speed reparametrization of the <harmonic oscillator equation>.
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