Mixed endpoint Green kernel for a first-derivative drift ($y^{\prime \prime }+y^{\prime }=f,y(0)=0,y^{\prime }(1)=0$)

Multiplying the equation for $v=y^{\prime }$ by the integrating factor $e^x$ gives $v(x)=-e^{-x}{\int }_x^1{e}^{t}f(t)dt$. Integrating once more and reversing the order of integration gives the Green's functionIt is continuous across $x=t$, its first derivative has jump $+1$, and it satisfies the mixed Dirichlet boundary condition and Neumann boundary condition. Both homogeneous solutions, $1$ and $e^{-x}$, are eliminated by these two conditions, giving uniqueness. For $f=e^{-x}$ the solution is $-x{e}^{-x}$.