In general, we use Ito's formula to solve linear stochastic differential equations. Consider for instance the geometric brownian motion:
$$dX_t = \alpha X_t dt + \beta X_t dW.$$
My question is:
How can I solve this a non-linear stochastic differential equation, like:
$$dX_t = \alpha X_t dt + \beta \sqrt{X_t} dW.$$
Can I still use Ito's lemma or do I need to transform the SDE? If I have to transform the SDE, then how and what is the general rule? Any help is appreciated.
For most SDE, I don't think there exists "general rule" that can apply in finding their analytic solutions. For some explicitly solvable cases you may be interested in this.