Consider the system
$$ \mathcal{L}u=f(x), x\geq0 \\ \ u(0)=\alpha,\ \ =\frac{du}{dx}(0)=\beta $$ We want to identify the Green's function for this problem, or in other way write the solution of $u$ in terms of a Green's function.
$\mathbf{My\ attempt}$
Let $v$ be in $C^2([0, \infty)$) such that $$\mathcal{L}v=f,\ v(0)=0=v'(0)$$, then for the above system the associted Green's function is defined to be the solution of $$\mathcal{L}G(t|\xi)=\delta(x- \xi),G(0|\xi)=0=G'(0|\xi)$$ and the solution of $v$ is given by $$v(x)=\int_0^1G(\xi|x)f(\xi)d \xi$$ Also for the adjoint Green's function we need $G^*(b)=\frac{dG^*}{dt}(b)=0$, since $$<G^*,\mathcal{L}G>-<G,\mathcal{L}^*G^*>=G^*(b)G'(b)-G(b)G^*(b)$$ where $G^*:=G^*(x|\eta),$ so $G^*(\xi|\eta)=G(\eta|\xi)$ and $x \in [0,b).$ I am not sure if my argument with $b$ correct or not, I needed somehow to have known value at a boundary in order to have the solution of the original problem.
I would apprectiate any help.