Differential Equations

Code	13928
Year	3
Semester	S1
ECTS Credits	6
Workload	TP(60H)
Scientific area	Mathematics
Entry requirements	NA
Learning outcomes	(i) To understand concepts and fundamental results from the theory of ordinary and partial differential equations; (ii) To use results from ordinary differential equations theory to analyse equations or systems of ordinary differential equations; (iii) To understand and to use some results from partial differential equations theory, with incidence in the wave, heat and Laplace equations; (iv) To analyse and understand mathematical proofs; (v) To communicate using mathematical language, written and orally.
Syllabus	1. Differential equations 1.1. Generalities and geometric interpretation 1.2. Differential equations with separable variables 1.3. Scalar linear differential equations 1.4. Exact differential equations 1.5. Differential equations of higher order 1.6. Method of undetermined coefficients 2. Linear differential equations 2.1. Linear differential equations in the plane 2.2. Exponential of matrices 2.3. Jordan canonical form 2.4. Flow of a linear differential equation 2.5. Non homogeneous linear differential equations 3. Non-linear differential equations in IRn 3.1. Flow of a non-linear differential equation 3.2. Existence and uniqueness of solution 3.3. Continuous dependence on initial conditions and parameters 3.4. Differentiability of the flow 3.5. Local stability 4. Partial differential equations 4.1. Linear equation and principle of superposition 4.2. Heat equation and Fourier method 4.3. Laplace equation 4.4. Wave equation and d’Alembert formula
Main Bibliography	- Arnold, V. (1974). Equações Diferenciais Ordinárias. Moscovo: Ed. Mir. - Braun, M. (1993). Differential Equations and Their Applications. Springer. - Chicone, C. (2006). Ordinary Differential Equations with Applications. Texts in Applied Mathematics, 34. (2ª edição). Springer. - Doering, C. I., & Lopes, A. O. (2016). Equações Diferenciais Ordinárias. Coleção Matemática Universitária. (6ª edição). IMPA. - Hirsch, M. W., Smale, S., & Devaney, R. L. (2013). Differential Equations, Dynamical Systems, and an Introduction to Chaos. (3ª edição). Elsevier Inc.
Language	Portuguese. Tutorial support is available in English.