Math 221 -- Things you need to know for exam 2

The exam covers chapter 2 and the first two sections of chapter 3.  However,
there are also some things from chapter 1 and some things done in class
(or in handouts) which you must know for the exam.  The following list
summarizes most of what you are responsible for.

1.  Systems of ODE's -- notation and terminology.  Know what an n-dimensional 
system of ordinary differential equations (ODE's) is.  Such a system can be
presented either by writing out n equations or by way of a single vector
equation.  Be conversant with both approaches!  Special kinds of systems:
linear, autonomous.  You should also know about the following special kinds
of linear systems: constant coefficient, homogeneous.

2.  Systems of ODE's -- solutions.  A solution to an n-dimensional system
is an n-tuple of functions (of the independent variable).  This can also be
expressed in vector form.  Obviously you should know how to check if a
purported solution really is a solution.  An n-dimensional system will normally
have an n-parameter family of solutions (described by the general solution,
which will have n constants in it).  If n initial conditions are in
addition specified (so that one has an initial value problem), one normally
expects there to be a unique solution.  A precise version of this
statement (without the ambiguous word "normally") is given by the existence
and uniqueness theorem (p. 167), which you should understand.  Understand
equilibrium solutions and how to find them.

3. Two-dimensional systems of ODE's -- graphical representation of solutions.
Since there are three variables, if we somehow display solutions in the
plane, one "dimension of information" will be lost.  Nevertheless it is 
very convenient to do so.  One approach is to plot what is happening in
the (x,y)-plane, if x and y are the names of the dependent variables.  One
can then see phase portraits, vector fields, and directions fields.  In
fact for vector fields, the independent variable information is captured.
Alternatively, one can graph x(t) or y(t).

4. Conversion of higher-order ODE's into systems.  An n-th order ODE can be 
converted into an n-dimensional system of ODE's.  (In a given situation, this
may or may not be a useful conversion to perform.)  Know what initial
conditions for the n-th order ODE look like, and be able to convert them
into initial conditions for the system.

5. Physical examples of second order ODE's: mass-spring system, falling bodies.

6. Euler's method for systems.

7. Qualitative analysis of systems via nullclines.

8. Complex numbers.  See the handout with some problems (and the solution
sheet for it).  You must understand everything on these sheets.

9. Matrices.  Understand what an n x k matrix is, and what the ij-th entry
of a matrix is.  (In both cases, the row number comes first.)  Know how
to add and multiply matrices (including when these operations are defined).
Be able to carry out routine manipulations of vectors and matrices.  (For
example, (AB)C = A(BC) for matrices, but AB is not equal to BA, in general.)
Know what the identity matrix is and why it is important.

10. Invertible matrices.  An n x n matrix A is invertible if there exists
an n x n matrix B such that AB = I and BA = I.  For a given matrix A,
there is at most one matrix B which will satisfy these equations.  (Know
how to prove this.)  This matrix B is called the inverse of A, and is 
denoted A^-1.  (That's A with the superscript -1.)  A square matrix is 
invertible if and only if its determinant is nonzero.  Know how to compute the 
determinant of a 2 x 2 matrix.  If A is invertible, the only vector x
satisfying Ax = 0 is 0, and conversely, if Ax = 0 has only the solution 
x = 0, then A is invertible.

11. How to solve a second order linear ODE which has constant coefficients
and is homogeneous.  Look for solutions of the form e^rt, where r is a
complex number.  This boils down to solving a quadratic equation in r.
If there are two distinct roots r1 and r2, then the general solution to
the ODE is c1 e^(r1 t) + c2 e^(r2 t).  You are done if r1 and r2 are real.
But it r1 and r1 are complex, you should also be able to go from the given
general solution to one involving no complex numbers.

12. Linear combinations and the linearity principle.  Understand thoroughly
what a linear combination of "things" is.  The application to systems of
ODE's (the linearity principle) is on p. 232.  It says that for a system of
the form dY/dt = AY, any linear combination of solutions is a solution.  This
assertion applies more generally to any linear homogeneous system of ODE's.
Know how to prove the linearity principle (as stated in the text).

13. Linear independence and its application to systems of ODE's.  
We say that x1,...,xn are linearly dependent if there exist numbers a1,...,an, 
not all zero, such that a1 x1 + ... + an xn = 0.  Otherwise, x1...,xn are 
linearly independent.  If Y1,...,Yn are linearly independent solutions to the 
n-dimensional system dY/dt = AY, then every solution to this system is
uniquely expressible as a linear combination of Y1,...,Yn.

14. Decoupled systems.

15. Know what eigenvalues, eigenvectors, and the characteristic polynomial
are.  Be able to use them to find straight-line solutions to systems and
more generally to find the general solution to the system.  

16. First-order ODE's -- explicit solution techniques for separable and
linear equations.  Don't forget this material!