I have an SDE that looks that Ornstein-Uhlenbeck SDE :
$$dX_t=(-\alpha +\beta X_t)dt+\sigma dB_t.$$
I know that $$d(X_te^{\beta t})=e^{-\beta t}dX_t-e^{\beta t}X_t\beta dt=e^{-\beta t}(-\alpha +\beta X_t)dt-e^{\beta t}\sigma dB_t+e^{\beta t}X_t\beta dt$$ $$=-\alpha e^{-\beta t} dt+\sigma e^{-\beta t}dB_t$$ does it work ?
Indeed using the integrating factor approach as here Solution to General Linear SDE for $dX_t = \big( a(t) X_t + b(t) \big) dt + \big( g(t) X_t + h(t) \big) dB_t,$ we get
\begin{align*} X_t = & X_0 e^{ \beta t}+ e^{ \beta t}\left( \int_0^t e^{ - \beta s}(-\alpha) \mathrm{d}s + \int_0^t e^{ - \beta s}\sigma \mathrm{d}B_s\right). \end{align*}