This paper deals with how the purely mathematical approach can be used to solve transient-state instability problems of dissolution-timescale reactive infiltration (DTRI) in fluid-saturated porous rocks. Three key steps involved in such an approach are: (1) to mathematically derive an analytical solution (known as the base solution or conventional solution) for a quasi-steady state problem of the dissolution timescale, which is viewed as a frozen state of the original transient-state instability problem; (2) to mathematically deduce a group of first-order perturbation partial-differential equations (PDEs) for the quasi-steady state problem; (3) to mathematically derive an analytical solution (known as the perturbation solution or unconventional solution) for this group of first-order perturbation PDEs. Because of difficulty in mathematically solving a transient-state instability problem of DTRI in general cases, only a special case, in which some nonlinear coupling between governing PDEs of the problem can be decoupled, is considered to illustrate these three key steps in this study. The related theoretical results demonstrated that the transient chemical dissolution front can become unstable in the DTRI system of large Zh numbers when the long wavelength perturbations are applied to the system. This new finding may lay the theoretical foundation for developing innovative technique to exploit shale gas resources in the deep Earth.