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Differential Equations

Objectives

The program deals with the study of first and second order differential equations, system of differential equations and partial differential equations. Some related topics will be studied in the end.

General characterization

Code

12968

Credits

6.0

Responsible teacher

José Maria Nunes de Almeida Gonçalves Gomes

Hours

Weekly - 4

Total - 56

Teaching language

Português

Prerequisites

Calculus I and II, and Linear Algebra and Analytical Geometry I.

Bibliography

The course has a support text which will be updated and made available periodically in the Support Documentation section in CLIP.

In addition to the support text, the following bibliography is recommended:

Ordinary Differential Equations and Partial Derivatives:

M. Braun. Differential Equations and their applications (4th edition). Springer-Verlag, 1993.

Elementary Differential Equations, with boundary value problems, C.H. Edwards and David E. Penney, Prentice Hall.

Fourier Analysis and Partial Differential Equations, ed. IMPA, Euclides project, 1977.

Teaching method

Theoretical-practical classes, with lectures, and solving exercise sessions under the supervision of the teacher. 

Evaluation method

Three 1-hour tests during school hours, evenly distributed throughout the semester and with a maximum mark of 20. The final grade is the arithmetic mean of the three tests. The student passes if the final grade is equal to or higher than 9.5.

Subject matter

1. First order ordinary differential equations. Existence and uniqueness of solution. Integrating Factor. Geometric
interpretation. Applications.
2. Second and higher order linear ordinary differential equations. Existence and uniqueness of solution. General
solution of the homogeneous equation. Methods to solve nonhomogeneous equations. Applications.
3. Systems of differential equations. Existence and uniqueness. Linear systems with constant coefficients.
Fundamental Matrices. Applications.
4. Introduction to the Sturm-Liouville theory. Examples.
5. Introduction to Fourier Series. Examples.
6. Partial differential equations. Examples. Characteristics.
7. Wave equation. Heat equation. Laplace equation. Separation of variables. Examples.
8. Introduction to the Calculus of Variations. Classical problems in the Calculus of Variations. Euler-Lagrange
Equations. Hamilton Principle.