The book presents advanced methods of integral calculus and optimization, the classical theory of ordinary and partial differential equations and systems of dynamical equations. It provides explicit solutions of linear and nonlinear differential equations, and implicit solutions with discrete approximations.The main changes of this second edition are: the addition of theoretical sections proving the existence and the unicity of the solutions for linear differential equations on real and complex spaces and for nonlinear differential equations defined by locally Lipschitz functions of the derivatives, as well as the approximations of nonlinear parabolic, elliptic, and hyperbolic equations with locally differentiable operators which allow to prove the existence of their solutions; furthermore, the behavior of the solutions of differential equations under small perturbations of the initial condition or of the differential operators is studied.Key FeaturesSurvey on integral calculus and differential equationsTechniques for the resolutions of the equations and equivalences between several classes of equationsSelf-contained and accessible to undergraduate and graduate students in mathematicsA large part devoted to the integral calculus with detailed solutionsIllustrations by examples and graphs of functionsExercises for each chapterApplications of the systems of differential equations to models of physics, geometry, population dynamics, demography, and biology

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