I'm self-learning SDEs and trying to understand how to solve SDEs using Ito's Lemma, so when given a general SDE, how can one choose the function to use? This is an example I found
$$ dX = \frac{1}{X} dt + \alpha X dB_t, X_0=x>0$$
Firstly, since there is a $dt$ and $dB_t$ terms, would we choose the bivariate Ito formula such that $$ df(t,X) = f(0,X_0) +\frac{\partial f(t,x)}{\partial t} + \frac{\partial f(t,x)}{\partial X_t} dX_t + \frac{\partial f^2(t,x)}{\partial X_t^2}d\langle X\rangle $$ I guess, since we know behaviour of BM, we should make the function in terms of $B_t$, because then we can simplify $d\langle B_t\rangle=dt$