= Mixed endpoint Green kernel for a first-derivative drift
{c}
{title2=$y''+y'=f,\quad y(0)=0,\ y'(1)=0$}
Multiplying the equation for $v=y'$ by the <integrating factor> $e^x$ gives $v(x)=-e^{-x}\int_x^1e^tf(t)\,dt$. Integrating once more and reversing the order of integration gives the <Green's function>
$$
G(x,t)=\begin{cases}1-e^t,&t\leq x,\\e^{t-x}-e^t,&x\leq t.\end{cases}
$$
It 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 $-xe^{-x}$.
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