Math 323 Fall 2005                                    last modified 11/21

This is an introduction to differential equations.  Half the course 
is ODEs, half PDEs.  Some theory is done, some solution methods, even 
a few derivations from physics.  You should know the mean value theorem, 
divergence theorem, the chain rule, linear algebra, and it would be best 
if you have solved a few de's already.  Most of these things will be
reviewed when first used. 

Insight into many of our equations can be gained by experimenting with 
the applets at http://www.math.cornell.edu/~bterrell. Also see the Notes 
on Differential Equations there, for background at the level of Math 293.

There will be two prelims in class, with opportunity to do corrections, 
and a final exam. Homework is due Friday for the topics covered by Wednesday.

ODE (Brauer and Nohel)

week
 1+2 8/26-9/2 Systems of ODE
        1.2: 3;  
        1.3: 1,2,3; 
        1.4: 3,8; 
        1.5: 1;
        1.6: 1,7,8; 
        1.7: 1,2;
        read handout on the Euler numerical method and do 1,2;

 3   9/6-9/9 Linear Systems
        2.1: 6,7; 
        2.3: 1,9,11,24;  
        2.4: 2,3,6;

 4   9/12-9/16 Linear Systems
        2.5: 3,4,7,25,27; 
        page 103: 1,14b;

 5   9/19-9/23 Linear Systems
        2.8: 3,4,5,6,9,13,19;

 6   9/26-9/30 Existence
        3.1: 1,2,5,6a,14;
        3.2: 3;
        3.3: read Theorem 3.4; 
        3.4: 3; read through Theorem 3.6; 

 7   10/3-10/7 Stability
        4.2: 1,3;  
        4.3: 17;
        5.1: 1;  
        5.2: 4; and show that the 0 solution of x''+(x^2)x'+x^3 = 0
        is stable. Numerically it looks asymptotically stable; can
        you show that?  what happens if x^2 is replaced by -x^2?





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PDE (Zachmanoglu and Thoe)

week
        Fall Break
 8   10/12-10/14 Well Posed Problems, Heat Equation derived
        I: read as needed for notation etc.
        VI: Explain all details of the divergence theorem in the case of the 
            5-dimensional ball and the 4-sphere which is its boundary.  You 
            may mimic the 3-ball,2-sphere case from a calculus book if you wish.
        Prelim 1 Friday in class covers through 3.4
 
 9  10/17-10/21 The Laplace Equation
        VI: 1.2, 1.4;
        VII: Derive equation (2.2) on page 174 by using the chain rule,
             2.3, 2.7a;

10  10/24-10/28 Laplace
        VI: 4.1b, 5.3;
        VII: 5.6, 12.3, 12.7;

11  10/31-11/4 Laplace, First Order Equations
        VII: 7.5, 9.4, 15.1; 
        Read handout on Euler and acoustic eqns: Exercise 1;

12  11/7-11/11 First Order, Wave Equations
        Euler handout: Exercises 2,3,4;
        III.5: Read sections 1-3 of the Lax paper;
        III.5: 5.1, 5.2;
        VIII: 1.1, 1.2, 1.6;

13  11/14-11/18 
        Lax paper: read jump conditions
        Prelim 2 Friday in class covers through 5.2

14  11/21-11-23 
        III.6 Traffic example: Use jump conditions from the Lax paper
              on equation III(6.6) (not problem 6.6) to find some traffic 
              conditions where the shock wave is stationary at x = 0, say 
              constants d(x,t) = c1 when x < 0, c2 when x > 0.
        Thanksgiving Break

15  11/28-12/2 The Wave Equation
        VIII: 6.1a, 6.5, 6.6;
        VIII.3-VIII.6: Let u(x,t) = sin(kx)cos(kt) in Omega = [0,pi].
                       1) Find what numbers k are allowed, so that 
                       u is a solution to the 1-dimensional wave equation
                       in Omega, 0 on the boundary, with initial
                       velocity 0.
                       2) Use some trig identity to write
                       u as a sum of traveling waves.
                       3) Calculate the energy integral of u and
                       confirm that it does not depend on t.
        VIII: 9.2;


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