Calculus I, Cálculus II and Linear Algebra

SPECIFIC LEARNING OBJECTIVES (PO a): - To understand the Linear operators and the principle of superposition for solving differential equations. - To solve elementary differential equations by separation of variables and other methods. - To understand the different application scopes of the differential equation to engineering and physics. - To distinguish between elliptic, hyperbolic and parabolic partial differential equations and which initial or boundary conditions are appropriate for them. - To understand how to apply separation of variables and the Fourier method to solve initial-boundary value problems for the equations of Mathematical Physics. - To understand the separation of variables technique, the role of the resulting eigenvalue problems and the principle of superposition to solve initial-boundary value problems for the equations of Mathematical Physics. - To understand when and how to use the method of characteristics to solve different cases of partial differential equations. GENERAL ABILITIES (PO a, g, k): - To understand the necessity of abstract thinking and formal mathematical proofs. - To acquire communicative skills in mathematics. - To acquire the ability to model real-world situations mathematically, with the aim of solving practical problems. - To improve problem-solving skills. (PO: a)

1. Introduction 1.1 Basic models; direction fields 1.2 Classification of differential equations 2. First Order Differential Equations 2.1 Linear equations; integrating factors 2.2 Separable equations 2.3 Exact equations 3. Second Order Linear Equations 3.1 Definitions and examples 3.2 Linear homogeneous equations 3.3 Homogeneous equations with constant coefficients 3.4 Inhomogeneous equations: undetermined coefficients 3.5 Variation of constants 4. Systems of First Order Linear Equations 4.1 Basic theory; higher-order equations 4.2 Explicit solutions of non-homogeneous linear systems 4.3 Planar linear systems 5. Nonlinear Systems and Stability 5.1 Planar nonlinear systems 5.2 Stability 5.3 Periodic solutions 5.4 Higher-dimensional systems 6. Partial Differential Equations: Introduction 6.1 Examples and physical derivation 6.2 Types of equations and data; well- vs ill-posed problems 7. Separation of Variables 7.1 Problem resolution by separation of variables 7.2 Fourier trigonometric series: basic properties 8. Boundary-value Problems 8.1 Sturm-Liouville problems 8.2 Self-adjoint operators and spectrum 8.3 Rayleigh¿s quotient 8.4 Generalized Fourier series 8.5 Multivariable Sturm-Liouville problems 9. Non-Homogeneous Problems 9.1 Shifting the data 9.2 Fredholm¿s alternative 9.3 Eigenfunction expansions

1.- Master classes. 2.- Problem classes. 3.- Partial controls. 4.- Final exam. 5.- Tutorials.