We　address　the　linear-mode　analysis　performed　near　an　equilibrium　configuration　in　the　fluid　rest　frame　with　a　dynamical　magnetic　field　perturbed　on　a　constant　configuration.　We　develop　a　simple　and　general　algorithm　for　an　analytic　solution　search　that　works　on　an　order-by-order　basis　in　the　derivative　expansion.　This　method　can　be　applied　to　general　sets　of　hydrodynamic　equations.　Applying　our　method　to　the　first-order　relativistic　magnetohydrodynamics,　we　demonstrate　that　the　method　finds　a　complete　set　of　solutions.　We　obtain　two　sets　of　analytic　solutions　for　the　four　and　two　coupled　modes　with　seven　dissipative　transport　coefficients.　The　former　set　has　been　missing　in　the　literature　for　a　long　time　due　to　the　difficulties　originating　from　coupled　degrees　of　freedom　and　strong　anisotropy　provided　by　a　magnetic　field.　The　newly　developed　method　resolves　these　difficulties.　We　also　find　that　the　small-momentum　expansions　of　the　solutions　break　down　when　the　momentum　direction　is　nearly　perpendicular　to　an　equilibrium　magnetic　field　due　to　the　presence　of　another　small　quantity,　that　is,　a　trigonometric　function　representing　the　anisotropy.　We　elaborate　on　the　angle　dependence　of　the　solutions　and　provide　alternative　series　representations　that　work　near　the　right　angle.　This　identifies　the　origin　of　a　discrepancy　found　in　recent　works.　Finally,　we　discuss　the　issues　of　causality　and　stability　based　on　our　analytic　solutions　and　recent　developments　in　the　literature.　　©　2024　American　Physical　Society.