Solution of recursive differential equation

I have to integrate: $$dX_{t} = \alpha X_{t-1}dt$$ But I don't know if this should be interpreted as $$\frac{1}{X_{t-1}}dX_{t} = \alpha dt$$ How can this $X_{t-1}$ be integrated w.r.t $dX_t$? Or how can it be integrated as $\alpha X_{t-1}dt$?

It looks to me that what you have here is a delay differential equation. I would consider dividing by $dt$ to write it as $$\frac{d}{dt}X_t = \alpha X_{t-1},$$ which is similar to the example considered here, at the end of the section 'The Characteristic Equation'. I think you then want to consider $X_t$ as a function of $t$, and integrate in $t$ to solve.

As in that example, try an ansatz of the form $X_t=Ae^{\lambda t}$. Substituting this into the equation gives a characteristic equation of the form $$\lambda =\alpha e^{-\lambda},$$ which can be solved for $\lambda$ in terms of the Lambert W function. Rearranging gives $\lambda e^{\lambda} =\alpha,$ with solution $\lambda = W(\alpha)$.

Then your original equation has solution $$X_t=Ae^{W(\alpha) t}.$$