Green function under dependent-variable rescaling ($L(qv)=rMv⟹G_L(x,s)=q(x)G_M(x,s)/r(s)$)

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 $Mv=f/r$, soThus 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 $Ly=x^2{y}^{\prime \prime }-2x{y}^{\prime }+(x^2+2)y$ on $[\pi /2,\pi ]$ with zero endpoint values, $q=x$ and $r=x^3$ reduce the equation to $Mv=v^{\prime \prime }+v$. 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 $L$.