I have the following problem which comes with the solution, but I am unable to obtain the solution... Any help would be greatly appreciated - I am preparing for finals :( Thanks a lot!
The SDE that I need to solve is $dX=-Xdt+e^{-t}dW$. The solution is $X(t)=(X_0+W(t))e^{-t}$
I have noticed that $$dW=e^{t}dX+e^{t}Xdt=d(e^tX)$$
So I have tried with $Z=e^tX$ and get $dZ=2e^t(dX+X)$, but I am stuck there now :/ (and I am not sure if that $dZ$ is correct either because we are not shown how the professor got the final result).
I think you were not quite right in calculating $dZ$. You had the right path though.
$$dZ=e^t X dt+e^t dX+e^t dt dX$$ $$dZ=e^t Xdt+e^t(-Xdt+e^{-t}dW)+0$$ $$dZ=(e^tX-e^tX)dt+dW$$ $$dZ=dW$$ $$Z=X_0+W(t)$$ $$X(t)=(X_0+W(t))e^{-t}$$