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In this paper, by using the characteristic system method, the kernel of a polynomial differential equation involving a derivation in <inline-formula> <math display="inline"> <semantics> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mi>n</mi> </msup> </mstyle> </semantics> </math> </inline-formula> is described by solving the Cauchy Problem for the corresponding first order system of PDEs. Moreover, the kernel representation has a special significance on the space of solutions to the corresponding system of PDEs. As very important applications, it has been established that the mathematical framework developed in this work can be used for the study of some second-order PDEs involving a finite set of derivations.