# 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?

• Check documentation page for RSolve. – Daniel Lichtblau Sep 17 '20 at 13:21
• @Daniel Lichtblau, RSolve does not give the correct answer. The solution does not match the original equation. The point is that the q-analog is not a recurrent equation. – dtn Sep 18 '20 at 5:39
• I may be wrong, but the fact is that the solutions do not match (even with q values close to 1) – dtn Sep 18 '20 at 8:04
• Run the following sequence to compare: ff0 = RSolveValue[{x[q*t] - x[t] == (q*t - t)*(-x[t]), x == 2}, x[t], t] ffq = ff /. q -> 101/100 N[Table[ffq, {t, 1., 2., .1}]] Table[2*Exp[1 - t], {t, 1., 2., .1}] The two tables give very close values. – Daniel Lichtblau Sep 18 '20 at 14:21