= Green function under dependent-variable rescaling
{title2=$L(qv)=rMv\implies G_L(x,s)=q(x)G_M(x,s)/r(s)$}
Suppose a linear ordinary <differential operator> satisfies $L(qv)=rMv$ on an interval, where the functions $q,r$ never vanish and homogeneous endpoint <boundary conditions> for $y=qv$ correspond to those used for $v$. If $G_M$ is a <Green's function> for $M$, then $Ly=f$ is equivalent to $Mv=f/r$, so
$$
y(x)=q(x)\int G_M(x,s)\frac{f(s)}{r(s)}\,ds.
$$
Thus the displayed kernel is the <Green's function> for the original, possibly non-normalized operator. The source factor belongs at $s$, while the dependent-variable factor belongs at $x$.
For $Ly=x^2y''-2xy'+(x^2+2)y$ on $[\pi/2,\pi]$ with zero endpoint values, $q=x$ and $r=x^3$ reduce the equation to $Mv=v''+v$. The kernel of $M$ is $\cos(x_<)\sin(x_>)$, where $x_<=\min(x,s)$ and $x_>=\max(x,s)$. Therefore $G_L(x,s)=x\cos(x_<)\sin(x_>)/s^3$. Its first derivative has jump $1/s^2$, as required by the leading coefficient $s^2$ of $L$.
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