# Proving uniqueness of ODE solution

The ODE is the following:

$$\begin{cases} u''(x) = 0,\\[6pt] u(0) = a, u'(0) = b \end{cases}$$

I need to prove that ODE is well-posed and so far I have proven both the existence of the solution and the stability, but I am not sure how to approach the uniqueness problem, assuming I don't know any particular theorems regarding the uniqueness proof.

My apologies if this is really trivial. I am not looking for an answer for this problem, but simply for a hint. Thanks.

Hint - suppose you had two (twice differentiable) solutions, $$u,v$$. Then the difference $$w = u-v$$ solves the same ODE and with zero initial data. Integrate from $$0$$ to $$x$$ for some arbitrary $$x$$, apply fundamental theorem of calculus and initial data...