Theorem: The inhomogenous boundary value problem has at most one solution if and only if the homogenous boundary value problem has only the trivial solution $u\equiv 0$.
where the inhomogenous BVP is: $Lu = f$ for $a<x<b$ and $B_1 u= \alpha_1$ at $x=a$ and $B_2 u = \alpha_2$ for $x=b$, and the homogenous BVP is the same system but where $f=\alpha_1 = \alpha_2=0.$
(Where $L$, $B_1$ and $B_2$ are linear differential operators).
This theorem was provided to us in class, but I'm not really sure why it's true? It was supposed deduced from simple facts like the superposition of homogeneous solutions is a homogeneous solution and the difference between homogeneous solutions is a homogeneous solution. However I don't really see how it follows?
Can someone please explain why this might be true? Thank you in advance!
You already have written it. The difference of two different inhomogeneous solutions would be a non-trivial homogeneous solution. If these do not exist, then there can only be at most one inhomogeneous solution.