I want to solve the following stochastic differential equation
$dX_{t}=X_{t}dt+dB_{t}$ where $B_{t}$ is standard Brownian motion.
The hint to this exersice is to "multiply by $e^{-t}$ and compare with $d(e^{-t}X_{t})$"
Now when I follow these instructions I obtain $e^{-t}dX_{t}=e^{-t}X_{t}dt+e^{-t}dB_{t}$ and by Ito lemma I get $d(e^{-t}X_{t})=e^{-t}dB_{t}$. Combining this we get $e^{-t}dX_{t}=e^{-t}X_{t}dt+e^{-t}dB_{t}=e^{-t}X_{t}dt+d(e^{-t}X_{t})$
which is the same as,
$d(e^{-t}X_{t})=e^{-t}dX_{t}-e^{-t}X_{t}dt$ (but this is what Ito lemma gives aswell)
hence,
$X_{t}=e^{t}X_{0}+\int_{0}^{t}e^{t-s}dB_{s}$
But the twist is that so does just applying Ito formula on $X_{t}$ with $g(t,x)=e^{-t}x$ to obtain $d(e^{-t}X_{t})$ and then taking the integral.
Have I misunderstood this hint or is it confusing? Why dont he ju say use the Ito formula? Since I use what I already know from Ito lemma, $d(e^{-t}X_{t})=e^{-t}dB_{t}$ to do the last step.
Maybe he makes one does it like this since it is hard to find integrating factors in general. Applying Ito lemma might be much harder in general.
Update! This guy seam to to have similar problems,
If $dX_{t} = X_{t}\,dt + \,dB_{t}$, why does $e^{- t}dX_{t} = e^{-t} X_{t} \,dt + e^{-t} \,dB_{t}$?