EM 223 - Mathematical Analysis III (Análise Matemática III) Introduction to differential equations: general classification, definition of solution and of boundary value problems. Ordinary differential equations of first order: the existence and uniqueness theorem; separable equations; homogeneous and reducible to homogeneous equations; linear equations (homogeneous and non homogeneous). Some problems modeled by first order equations: problems in mechanics, population dynamics and orthogonal trajectories. Exact equations and integrating factors. Non linear equations reducible to linear ones: the Bernoulli and the Riccatti equations. Ordinary second order differential equations reducible to first order equations: the two special cases of missing the independent variable or the dependent variable. Linear equations of order greater than one: general theory of homogeneous and non homogeneous linear nth order equations. Existence and uniqueness theorem. General solution for homogeneous linear equations with constant coefficients. Linear non homogeneous equations: the variation of parameters method and the annihilator method. Systems of first order linear equations: introduction and its relation with an nth order linear differential equation. Some examples. Basic theory of systems of first order linear equations. Homogeneous linear equations with constant coefficients. Real or complex single eigenvalues case and repeated eigenvalues case. Fundamental matrices. The method of variation of parameters for non homogeneous systems The Laplace transform: definition and existence conditions. Laplace transform of some basic functions using the definition. Main properties of Laplace transform: first and second translation theorems and the transform of the derivative. Inverse Laplace transform. Solution of initial value problems and of differential equations with discontinuous forcing functions, using the Laplace transform. Impulse functions and Dirac δ-function. The convolution theorem.. <<<<<<<<<<  BACK  <<<<<<<<<<