 Bounded Green kernel for a radial Euler operator

For $Ly=(x^2{y}^{\prime }{)}^{\prime }-2y$ on $x>0$, choose solutions bounded at zero and infinity. The homogeneous Cauchy-Euler equation has basis $x,x^{-2}$. Joining the left-regular and right-regular solutions continuously and imposing $s^2[G_x{]}_{s^{-}}^{s^{+}}=1$ gives the displayed Green's function with ordinary integration measure $ds$. For bounded forcing,The integrals converge and give the unique bounded solution: no nonzero homogeneous combination is bounded at both ends. If $0\le f\le M$, negativity of the kernel gives $-M/2\le y\le 0$. Differentiation cancels the two moving-endpoint terms, leavingwhich implies $|y^{\prime }(x)|\le M/(3x)$. For constant forcing $f=M$, the bounded solution is $-M/2$; the endpoint conditions are boundedness, rather than zero endpoint values.