This is a problem from a lecture note.
The differential equation is like this $d \mathrm{X}(t)=\alpha \mathrm{X}(t) d t+\sigma d \mathrm{B}(t), \quad \mathrm{X}(0)=x_{0}$
This is a picture of wrong way to solve it! Here we assume $X(t)=f(t,B(t))$. And use Ito's Lemma to get $dX(t)$ to match the RHS of differential equation.
The correct answer is by multiplying $e^{-\alpha t}$ on both sides and get $\mathrm{X}(t)=e^{\alpha t} x_{0}+\int_{0}^{t} \sigma e^{\alpha(t-s)} d \mathrm{B}(s)$
I am curious why the wrong way does not work. Specifically, what is $dX/dB$ and $dX/dt$ in the correct answer.