Syllabus

 

Math 319

Spring 2005

 

 

 

Marshall Slemrod

Email: slemrod@math.wisc.edu

Office: 523 Van Vleck Hall

Office hours: 11:00 – 12:00 Tuesday and Thursday

 



Text: Boyce & DiPrima, Elementary Differential Equations and Boundary Value Problems, 8th edition


Grading: Three in class quizzes (20% each), Final exam (40%)

 

Outline of Subject Matter:

 

1.  Introduction: definition of an ODE, basic problems (IVP and BVP), examples
 
2.  First order equations (1-1/2 weeks)
     a)   linear: homogeneous and inhomogeneous
     b)   nonlinear: separable
 
3.  a)   direction fields (1-1/2 - 2 weeks)
     b)   the basic existence and uniqueness theorem (for first order equations)
     c)    the Euler scheme
     d)   other numerical methods
 
4.  Second order linear equations with constant coefficients (1-1/2 - 2 weeks)
     a)   homogeneous case
     b)   inhomogeneous equations via methods of annihilators and variation of parameters
     c)    remarks on higher order equations, linear independence, and the Wronskian
     d)   applications to forced oscillation problems, effect of resonances
 
5.  Laplace transform (2 weeks)
     a)   definition and elementary properties
     b)   application to constant coefficient linear equations
     c)    discontinuous forcing terms
 
6.  First order systems (1-1/2 - 2 weeks)
     a)   conversion of 2nd and higher order equations to systems (focusing on
           systems in the plane and simple cases in 3 dimensions)
     b)   discussion of algebraic properties of vectors in and matrices on the plane
           and 3 dimensional space.  Also differentiation of vector and matrix functions
     c)    solution of linear constant coefficient systems
 
7.  Two dimensional systems and the phase plane (2-3 weeks)
     a)   classification of (equilibria for) linear systems
     b)   qualitative behavior of nonlinear systems: classification of equilibria; stability
     c)    applications, e.g. to the pendulum, population models
 
8.  Boundary value problems (2-3 weeks)
     a)   physical origins via separation of valuables from PDE
     b)   Fourier expansions
     c)    eigenvalue problems
     d)   More general expansion methods
 
9.  More on systems (time permitting)
(a)   qualitative behavior in the phase plane: limit cycles, heteroclinics, homoclinics, etc.; the Poincaré- 
      Bendixson Theorem.
     (b)  the dependence of equations on parameters; bifurcation.
     (c)  chaotic solutions.
 
10.  Series methods (time permitting)




Homework:



Section
 Page
 Problems
2.1
 39
 1, 3, 5, 7, 9, 13, 15, 17
2.2
 47
 3, 4, 10, 18, 23
1.1
 7
 1, 2, 3, 4
2.7
 108
 1, 2, 3
2.8
 117
 1, 2, 3, 4
3.1
 142
 1, 2, 3, 4, 5, 6, 11, 13, 22
3.2
 151
 3, 6, 9, 11, 14, 23, 25
3.3
 158
 1, 2, 8, 9, 13, 15, 19, 22
3.4
 164
 2, 4, 11, 13, 17, 18, 19
3.5
 172
 1, 3, 5, 11, 13, 25, 27
3.6
 184
 1, 3, 5, 7, 9, 10, 14, 15
3.7
 190
 3, 6, 7, 8, 14, 17
3.9
 214
 5
4.1
 222
 3, 5, 7, 9, 12
4.2
 230
 3, 5, 10, 11, 12, 13, 14, 15, 16, 17, 18
6.1
 312
 1, 3, 7, 9
6.2
 322
 5, 11, 12, 13, 14, 21, 22, 23
6.3
 329
 3, 5, 9, 11, 15, 20, 21, 24, 26
6.4
 337
 1, 2, 3, 4, 5, 9, 10
6.5
 344
 1, 2, 3, 5, 10, 11
6.6
 351
 4, 5, 6, 7, 8, 9, 10, 11
7.1
 360
 3, 4, 10, 11
7.2
 372
 1, 3, 8, 10, 14, 18, 24, 27
7.3
 383
 1, 2, 3, 4, 8, 9, 13, 14, 15, 16
7.4
 389
 1, 3, 5, 6
7.5
 398
 3, 5, 7, 16, 17, 24, 25
7.6
 410
 3, 5
7.7
 420
 2, 3, 9, 10
7.8
 428
 1, 2, 5
7.9
 439
 2, 3, 5
9.1
 492
 3, 5, 7, 14, 15
9.2
 501
 1, 2, 7, 8
9.3
 511
 5, 6, 7
10.1
 575
 3, 4, 7, 8, 11, 15
10.2
 585
 2, 3, 4, 6, 8, 14
10.3
 592
 2, 3, 11, 12, 13
9.7
 556
 1, 3, 8



 

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