Topics:

1.

SYLLABUS MASTER'S/PHD QUALIFYING EXAM

ORDINARY DIFFERENTIAL EQUATIONS

Elementary solution techniques (e.g. Math 316).

May 1996

2. Existence, uniqueness, and continuation of solutions and continuity with respect to parameters.

3.

4.

5.

6.

References:

Linear systems with constant and periodic coefficients.

Basic ideas of stability theory including the theory of almost linear systems and the theory of Liapunov functions.

Two-dimensional system. Phase plane portraits. Energy method; Poincare­Bendixon theory.

Regular perturbation theory and 1st order averaging theory.

1. V. Arnold, Ordinary Differential Equations. 11

2. W.E. Boyce and R.C. DiPrima, Elementary Differential Equations. 11

3. F. Brauer and J. Noel, "Qualitative Theory of Ordinary Differential Equations. 11

4. E. Coddington and N. Levinson, Theory of Ordinary Differential Equations. 11

5. J. Hale, "Ordinary Differential Equations. 11

6. M. Hirsch and S. Smale, "Differential Equations, Dynamical Systems, and Linear Algebra."

7. W. Hurewicz, "Lectures on Ordinary Differential Equations. 11

8. D. Jordan and P. Smith, "Nonlinear Ordinary Differential Equations. 11

9. L. Perko, "Differential Equations and Dynamical Systems. 11

.lV. ... F:--'Velhtti:Sr,' ~·l.Voniin-eariJiifetentirlii£i[amions anti vynarrUc'di 3ysr:erhs ...

Please refer questions concerning this syllabus to ellison@m.ath

SYLLABUS MASTER'S/PHD QUALIFYING EXAM

ORDINARY DIFFERENTIAL EQUATIONS

Topics: May 1996

L;.

1. Elementary solution techniques (e.g. Math 316).

2. Existence, uniqueness, and continuation of solutions and continuity with respect to parameters.

3. Linear systems with constant and periodic coefficients.

4. Basic ideas of stability theory including the theory of almost linear systems and the theory of Liapunov functions.

5. Two-dimensional system. Phase plane portraits. Energy method; Poincare-TLnlinru.x6hmtmnr. ___________ _ _______ ------------· ------- . -------

1. V. Arnold, Ordinary Differential Equations. 11

3.

4.

5.

6.

7.

8.

9.

10.

F. Brauer and J. Noel, "Qualitative Theory of Ordinary Differential Equations. 11

F. C:nililinat.nn !'!nil N T .~vin!':nn Tht>f)7"'\J nf nrrlinrt7"'\J Difforonfirtl Ti'n71rtfinnc;, II -· -------o---- ---- -·· - ................................ , -. ................. J ""I _. .......... ., • .., .......... J ~""11 ...... _':" ..... .,., ........ ., ........ "1......,'""':."""''-''"''-"•

J. Hale, "Ordinary Differential Equations. 11

M. Hirsch and S. Smale, "Differential Equations, Dynamical Systems, and Linear Algebra."

W. Hurewicz, "Lectures on Ordinary Differential Equations. 11

D. Jordan and P. Smjth. "Nonlinear Ordi7Jarv,!Jiffenmtin~ Eauation.<:."

L. Perko, "Differential Equations and Dynamical Systems. 11

F. Verhulst, "Nonlinear Differential Equations and Dynamical Systems. 11

Please refer questions concerning this syllabus to ellison@math