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This textbook presents derivations and analytical solution methods for the basic linear partial differential equations that model transport, heat, diffusion, waves, vibration, and steady-state equilibrium (Laplace and Poisson equations). General theorems about solution properties (such as uniqueness and stability) are presented. Solution methods include Dirichlet's principle, separation of variables, Fourier series, and Green's functions. The book includes 61 problems with detailed solutions. The book is suitable for students of undergraduate mathematics, engineering, and science.