 Solution

The Itô product rule with a deterministic integrating factor givesIntegrating from zero provesBoth 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 $e^{2t}{\int }_0^t{e}^{-2s}ds=(e^{2t}-1)/2$. The positive drift sign gives an unstable linear diffusion rather than a mean-reverting one.