= Solution
The <Itô product rule> with a deterministic <integrating factor> gives
$$
d(e^{-t}X_t)=e^{-t}dX_t-e^{-t}X_tdt=e^{-t}dB_t.
$$
Integrating from zero proves
$$
\boxed{X_t=e^t\int_0^t e^{-s}dB_s.}
$$
Both coefficients are globally <Lipschitz>, so part (a) proves that this is the pathwise unique solution. Under a measure where $B$ is a <Brownian motion>, the solution is centered <Gaussian> with variance $e^{2t}\int_0^te^{-2s}ds=(e^{2t}-1)/2$. The positive drift sign gives an unstable linear diffusion rather than a mean-reverting one.
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