I have to solve the following SDE:
$$dX_t=X_t dt+2W_tdW_t$$
Let $Y_t=X_t e^{-t}$. By Ito formula we have:
$$dY_t=-X_te^{-t}dt+e^{-t}(X_t dt+2W_tdW_t)=2e^{-t}W_tdW_t$$
Thus
$$Y_t=Y_0+2\int_0^te^{-s}W_sdW_s$$
Going back to $X_t$
$$X_t=e^{t}Y_0+2\int_0^te^{t-s}W_sdW_s$$
Is this correct?