Asymmetry of a class of Mellin transforms via bounded solutions

We introduce a family of parametrized non-homogeneous linear complex differential equations on $[1,\infty)$, depending on a complex parameter $s$ in the critical strip. We identify sufficient conditions on the non-homogeneous term that induce a structural asymmetry between the solutions corresponding to the parameters $s$ and $1-s$. More precisely, if both solutions with initial value $1$ are bounded on $[1,\infty)$, then necessarily $\Re(s)=\tfrac12$. The initial condition associated with the unique bounded solution corresponding to a parameter $s$ represents a zero of the Mellin transform associated with the non-homogeneous term at the point $s$. Finally, we obtain a real-valued functional formula that enables us to locate these zeros on the critical line.