This paper is devoted to the analytical study of the non-homogeneous functional differential equation y'(x)=ay(qx)+by(x)+1/x, where q is a constant in (0,1) and where a and b are two non-zero real or complex numbers. This equation represents a q-perturbation of the first-order ODE y'(x)=by(x)+1/x, that admits an irregular singular point at infinity in the complex plane. First, combing hypergeometric-type and q-hypergeometric-type power series yields solutions at zero and infinity. Next, using the Laplace transform allows us to convert this equation into a first-order linear q-difference equation. In this way, we express the above-mentioned power series-type solutions in terms of Laplace integrals involving the Jacobi theta functions. Finally, we get the connection formula between solutions at zero and infinity, which plays a crucial role in determining the asymptotic behavior of solutions at infinity.