differential equations – Solve ODE transformed into q-difference in Mathematica


Let’s take a simple differential equation:

$frac{dx}{dt} = -x(t)$

Suppose we want to do a q-transform. To do this, we will replace the time derivative $frac{d}{dt}$ with the q-derivative (also known as the Jackson’s derivative: q-derivative).

$frac{x(q cdot t) – x(t)}{q cdot t – t} = -x(t)$

Question: how to solve the resulting equation in Mathematica?