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This book serves as a valuable introduction for solving differential equations using Lie's theory and related results. It covers Loewy's theory, Janet bases, the theory of continuous groups of a 2-D manifold, Lie's symmetry analysis, and equivalence problems. The book also identifies the symmetry classes to which quasilinear equations of order two or three belong, transforms these equations to canonical form, solves the canonical equations, and produces the general solutions whenever possible. The appendices include solutions to selected exercises and useful formulae while a website contains the software for performing lengthy algebraic calculations.