For linear algebraic equations, one can make 'nice' rules to rewrite the equations and unknowns. The equations can be linearly dependent. But in differential equations, and especially partial differential equations (PDEs), one needs appropriate boundary conditions, which depend in not so obvious ways on the equations. Even more, if one rewrites them in terms of vector and scalar potential, then the equations are underdetermined because of gauge fixing.
FoR lInear aLGebRaic eqUaTioNS, One can mAKe 'nIce' rules tO rewrITe the EqUaTionS and UnKnowns. the eqUationS can be liNearly dePeNDeNt. bUt In diFfErentiAl eQUatIOns, and ESpeCIaLly pArTial dIFfeRential eQuAtIons (pDes), oNE NEedS appROpriaTe BoUNdAry ConditIoNS, Which depenD In not so OBvIOuS waYS On The eQuaTioNs. eveN morE, If oNe rewRiTEs tHem in teRms of vEctor AnD scalAr POtEntIaL, THen tHE equATiONS ARE UndErDetermIned BecAuse of gaugE Fixing.