Can $e^{t(X+Y)}=e^{tX}O_t$ be solved for $O_t$, when the operators $X$ and $Y$ don't commute?

Question

What is the exact solution $O_t$ of $e^{t(X+Y)}=e^{tX}O_t$, when $X$ and $Y$ are non-commuting operators on a Hilbert space? All I found is the Baker–Campbell–Hausdorff formula, but it gives the solution $Z$ of $e^Xe^Y=e^Z$.