1st order q-differential equations

Question

Can we solve 1st order q-differential equations using the usual methods of 1st order differential equations? For example, can we use integration factor method to solve this q-differential equations?

$$\text D_qy(x)=a(x)y(qx)+b(x)$$

Answer 1

Here is how to get a functional equation since you asked. Starting with a direct reference from q-Derivative from Wolfram MathWorld:

Therefore:

$$\text D_q y(x)=a(x)y(qx)+b(x)=\frac{y(qx)-y(x)}{x(q-1)}=a(x) y(qx)+b(x)$$

which works for $q\ne 1$ since:

$$\lim_{q\to1}\text D_q y(x)= \lim_{q\to1} \frac{y(qx)-y(x)}{x(q-1)} =\lim_{q\to0}\frac{y(x+q)-y(x)}{q}=y’(x)= \text D y(x)=\frac{dy(x)}{dx}$$

but this limit is not needed when the q-Derivative is taken. Therefore we have our functional equation, rewritten:

$$\boxed{\text D_q y(x)=a(x)y(qx)+b(x)\iff\frac{y(qx)-y(x)}{x(q-1)}=a(x) y(qx)+b(x)}$$

Here is another form which also gives a recursive solution:

$$\text D_q y(x)=a(x)y(qx)+b(x)\iff y(qx)-y(x)=a(x) y(qx)x(q-1)+b(x)x(q-1)\\\implies y(x)=y(qx)(a(x)x(1-q)+1)-b(x)x(q-1) $$

Please correct me and give me feedback!