1. Differential equations of first order.
1.1. Definitions and examples.
1.2. Elementary methods of resolution.
2. Higher order differential equations.
2.1. Linear differential equations of order n with constant coefficients.
2.2. Equations with variable coefficients: order reduction and equidimensional equations.
2.3. Relation between systems and linear equations.
3. Laplace transform.
3.1. Definition and properties.
3.2. Transforming and back-transforming.
3.3. Application to the resolution of linear equations and systems.
4. Method of separation of variables.
4.1. Initial and boundary problems. Examples of partial differential equations from Mathematical Physics.
4.2. Different kinds of equations and data.
4.3. Odd, even and periodic extensions of a function. Trigonometric Fourier series.
4.4. Resolution of equations by separation of variables and Fourier series.
4.5. Complex form of Fourier series.
5. Sturm-Liouville problems.
5.1. Sturm-Liouville problems and theorem.
5.2. Rayleigh's quotient. Minimization theorem.
5.3. Resolution of equations by separation of variables and generalized Fourier series.
5.4. Sturm-Liouville problems in several variables.