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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.

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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...

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