Solution
= Solution
The <Itô product rule> for $e^{\lambda t}X_t$ gives
$$
d(e^{\lambda t}X_t)=e^{\lambda t}\,dB_t.
$$
Therefore the <Ornstein-Uhlenbeck process> has the explicit form
$$
X_t=xe^{-\lambda t}
+\int_0^te^{-\lambda(t-s)}\,dB_s.
$$
= Solution
The <Itô product rule> for $e^{\lambda t}X_t$ gives
$$
d(e^{\lambda t}X_t)=e^{\lambda t}\,dB_t.
$$
Therefore the <Ornstein-Uhlenbeck process> has the explicit form
$$
X_t=xe^{-\lambda t}
+\int_0^te^{-\lambda(t-s)}\,dB_s.
$$