# The boundary value problem has a unique solution if and only if the corresponding homogeneous system has a unique solution

I want to prove that the following claims are equivalent:

1) The boundary value problem $$Ly = f, By = c$$ has a unique solution for all $$f \in \mathcal{C}[a,b]$$ and $$c \in \mathbb{C}^2$$,

2) $$Ly = 0, By = c$$ has a unique solution for all $$c \in \mathbb{C}^2$$.

I'm not sure where to start really. One idea I had was to assume that the claim in (1) is not true, so that there are two solutions $$y$$ and $$\tilde{y}$$, and then $$L(y-\tilde{y}) = Ly - L\tilde{y} = f-f = 0$$, but $$y \neq \tilde{y}$$, and since a homogeneous system $$Ly = 0$$ always have at least the trivial solution $$y = 0$$, this would imply that $$Ly = 0, By=c$$ has at least two distinct solutions. However, I'm not sure if it is even true that the trivial solution $$y=0$$ is a solution to $$Ly=0, By=c$$.

Any help?