Notation: the endpoints of the intervals are separated by ";" (instead of ",") -- e.g. $\ [0;1].$
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Given a function $\ C^{\infty}$-function $\ f:\mathbb R\to\mathbb R\ $ such that $$ \forall_{x\in\mathbb R}\quad f(x+1)-f(x)\ =\ f'(x) $$ all derivatives are also solutions: $$ \forall_{n\in\mathbb N}\,\forall_{x\in\mathbb R}\quad f^{(n)}(x+1)-f^{(n)}(x)\ =\ f^{(n+1)}(x) $$
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Given a function $\ C^{\infty}$-function $\ f_0:[0;1]\to\mathbb R\ $ such that $$ f_0(1)\ =\ f_0(0)\ + f_0'(0) $$ (there is a continuum of such different functions $\ f_0)$, we have (recursively) functions $\ f_n:[n;\,n+1]\to\mathbb R\ $ defined as follows:
$$ \forall_{n\in\mathbb R}\,\forall_{x\in[n;\,n+1]}\quad f_n(x)\ :=\ f_{n-1}(x-1)+f_{n-1}'(x-1) $$
(we work here with the one-sided derivatives at the integer points) hence
$$ \forall_{n\in\mathbb R}\,\forall_{x\in[n;\,n+1]}\quad f'_n(x)\ :=\ f'_{n-1}(x-1)+f_{n-1}''(x-1) $$
Then, by induction, $$ f_n(n)\,\ =\ f_{n-1}(n-1)+f_{n-1}'(n-1)\ =\,\ f_{n-1}(n) $$ and $$ f'_n(n)\,\ =\ f'_{n-1}(n-1)+f_{n-1}''(n-1)\ =\,\ f'_{n-1}(n), $$ and the same for the higher derivatives. Thus, there is a unique $C^{(\infty)}$-function $\ f:[0;\infty]\to\mathbb R\ $ that extends all $\ f_n,\ $ and it satisfies the OT's differential equation.
Clearly, a similar construction exists for arbitrary half-line $\ [a;\infty]\to\mathbb R;\ $ we may even simply define $\ g_a:[a;\infty]\to\mathbb R\ $ by $\ g(x):=f_0(x-a),\ $ where $\ f_0\ $ would be as above, etc.
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It's easy to see that the extension of zero function $\ \theta:[0;1]\to\mathbb R $ over $\ [0;\infty],\ $ that safisfies the OT's equation, is the zero function.
Also, any such extension $\ \Theta\ $ of $\ \theta,\ $ over any interval $\ (-1;1]\ $ for $\ a<0,\ $ must satisfy $$ \forall_{x\in[-1;0]}\quad \Theta(x)+\Theta'(x)\ =\ 0 $$ hence the extension would be of the form $\ C\cdot\exp(-x)\ $ hence $\ C=0.\ $ Thus the extension is the zero function. And step by step, any such extension in the negative direction must be the zero function. In particular, there is exactly one extension of $\ \theta\ $ onto $\ \mathbb R,\ $ namely the zero function.
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Now, it follows easily, that any OT's extension in the negative direction, if it exists, is unique. However, I have not addressed the question of the existence of the extensions in the negative direction.