Math 118A - Differential Equations

Schedule and Homework Assignments


The following is a rough projection of we will cover during the Spring Quarter. The schedule may change, and will be updated from time to time. See the
hints for downloading if you have problems accessing the archived homework assignments.

Week 1:
Introduction, explicit solution of a linear problem, basic concepts, linearization.

Homework Assignment due Friday, January 16:
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Week 2:
Classification of critical points, linear equations, matrix exponentials.

Homework Assignment due Friday, January 23:
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Week 3:
Norms, convergence of sequences in Banach spaces, contraction mapping theorem, Picard iterations, existence and uniqueness of solutions.

Homework Assignment due Friday, January 30:
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Week 4:
Existence and uniqueness of solutions (continued); stable and unstable manifolds.

Homework Assignment due Monday, February 9:
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Week 5:
Orbital derivative, first integrals, conservative systems, Hamiltonian system as an example, Liouville's theorem, potential and kinetic energy.

Week 6:
Verhulst, Chapter 4.
Midterm Exam on Wednesday, February 11.

Midterm Exam:
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Week 7:
Verhulst, Chapter 5.

Homework Assignment due Monday, February 16:

  1. Verhulst, Problem 4.3

  2. Verhulst, Problem 4.6

  3. Verhulst, Problem 4.8

    Note: The definition of the omega limit set given in the book is not the usual one. There is a subtle difference which will be discussed in class. However, the answer for the problem as stated is identical for the two definitions.

    Additional question: How does the answer to part (c) change if you consider complete orbits rather than positive orbits? Is the result still the same for the two definitions?

  4. Mathematica Problem:

    Define a Mathematica function to compute the divergence of a vector field. Then test several vector fields (take examples from Verhulst, or make up your own) in two dimensions to see if they satisfy Bendixon's criterion. Plot the vector field as well as some orbits of the associated differential equation for at least one example of each of the following cases:

    • A volume preserving flow.
    • A flow where the divergence of the vector field does not change sign in the phase space, or some reasonable subset of the phase space.
    • A flow where the divergence of the vector field does not change sign.


Week 8:
Verhulst, Chapter 6.

Homework Assignment due Monday, February 9:
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Week 9:
Verhulst, Chapter 6 (finish). Chapter 7 contains some additional material for concepts covered before. Reading material, in particular the examples and exercises.

Homework Assignment due Monday, March 9:

  1. Verhulst, Problem 5.1. Use Mathematica as much as possible for your symbolic manipulations.

  2. Verhulst, Problem 6.4

  3. Verhulst, Problem 6.7


Week 10:
Verhulst, Chapter 8.

Final Exam:
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Last modified: 1998/03/23
Marcel Oliver (oliver@math.uci.edu)
The Math 118 Home Page can be found at http://www.math.uci.edu/~moliver/math118/math118.html.