Differential Equations


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





Responsible teacher

Fábio Augusto da Costa Carvalho Chalub


Weekly - 4

Total - 56

Teaching language



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


We will follow the textbook in Portuguese available at



Teaching method

Classes consist on two different aspects: an oral explanation which is illustrated by examples and the resolution, by the students, of proposed exercises. Most results are proven. 

Evaluation method

Presence in class of optional.

Students are evaluted by two tests (T1 and T2), evaluated from 0 to 20; it is required minimum grade of 7 in the second test.

If T2<7, then MF=min(9,(T1+T2)/2); otherwise MF=(T1+T2)/2.

The final grade consists in the rounding of MF to the nearest integer (n.5 is rounded to n+1).

For students that to not obtain approval in tests ("continuous evaluation") it is possible to do a final exam ("recurso").

Subject matter

1. First order differential equations. Exact differentials. Integrating factors. Separation of variables. Homogeneous equations. Linear equations. Qualitative theory.

2. Second order differential equatons. Linear equations and Euler equation. Variation of constants.

3. Solution in series.

4. Higher order linear equations.

5. Systems of linear equation with constant coefficients. Differential equations in polar coordinates. Linearization of non-linear systens near equilibria.

6. Laplace transform and its use in differential equations. Dirac delta.

7. Partial differential equations. Heat, wave and Laplace equations. Laplacian in spherical coordinates.

8. Fourier series and transfors: applications do differential equations.

9. Introduction to variational calculus. The law of sinus (Snell-Descartes). Catenary. The principle of minimum action. Euler-Lagrange equation and the lagrangean. Brachistochone curve.