Mixed Green function for an Euler operator with exponents one and two ($Lz=x^2{z}^{\prime \prime }-2x{z}^{\prime }+2z,z(1)=z^{\prime }(2)=0$)

variation of parameters with the fundamental solutions $x,x^2$ givesThe kernel is continuous across $x=t$ and its first derivative jumps by $t^{-2}$, which gives unit in a ring source after multiplication by the leading coefficient. It satisfies both endpoint conditions. The homogeneous solution $Ax+B{x}^2$ satisfies them only when $A=B=0$, proving uniqueness.