Explicit solution of the Ornstein-Uhlenbeck equation

Solve the Ornstein-Uhlenbeck SDE $dX_t=\theta(\mu-X_t)\,dt+\sigma\,dW_t$ with $\theta>0$ and starting value $X_0$, and describe the law of $X_t$.

$X_t=\mu+(X_0-\mu)e^{-\theta t}+\sigma\int_0^te^{-\theta(t-s)}dW_s$. Multiply by the integrating factor $e^{\theta t}$; then $d(e^{\theta t}X_t)=\theta\mu e^{\theta t}dt+\sigma e^{\theta t}dW_t$ and integrate. $X_t$ is Gaussian with mean $\mu+(X_0-\mu)e^{-\theta t}$ and variance $\tfrac{\sigma^2}{2\theta}(1-e^{-2\theta t})$, tending to $N(\mu,\sigma^2/2\theta)$.