I am attempting to solve the Vasicek model SDE (using Wikipedia parametrisation):
$$ dr_t = a(b-r_t)dt + \sigma dW_t $$
Every solution is proceeding to multiply both sides of the equation by the integrating factor $e^{at}$ (akin to solving linear ODEs). After multiplication and rearrangement we get the following equation:
$$ e^{at}dr_t + e^{at}ar_tdt= e^{at}(abdt + \sigma dW_t) $$ Now the left hand side is apparently equal to $d(e^{at}r_t)$. How is that exactly the case?
Is it by Ito product rule? If so what is $X(t)$ and $Y(t)$?
Is it by Ito's lemma but then what is the $f(x,t)$