Hi I think i might have spotted an errata in my SDE textbook but I don't feel confident enough to confirm it. The question is compute $$ dX_t = X_tW_tdW_t $$ $W_t$ being the wiener process.
The textbook's answer is $$ X_t = exp( W_t^ 2 /2 - t/2) $$ which is obviously wrong if we differentiate using Ito's formula (unless i am missing something).
In any case, I am wondering if there is a method to solve this SDE systematically. I have tried integration factor method, exact method, inspection and I still can't find the solution. Any input appreciated.