Dear 293; 9/23/96 (two days after the max of -d(daylight)/d(day))
Guide to Chapter 3 reading
^^^^^^^^^^^^^^^^^^^^^^^^^^
Chapter 3 is about equations of the form: A(t)x'' + B(t)x' + C(t)x = F(t) (1)
where x=x(t) and x' and x'' are the first
and second derivatives of x w.r.t. t,
respectively. The general goal is to find
an x(t) that satisfies this equation for
given functions A(t), B(t), C(t) and F(t).
We, here in math 293 in the fall of 1996,
are only going to consider some special
cases all of which are of the form Ax'' + Bx' + Cx = D sin(omega t). (2)
That is, in (1) we are only considering
functions A(t) = A = constant
B(t) = B = constant
C(t) = C = constant
F(t) = D sin (omega t)
with D = constant
and omega = constant
As you read the text you should keep this in mind. Then, later, if you are
mathematically inclined and interested you can go back and look at things in
more depth.
We are especially going to emphasize a
few special cases of eqn (2). These are x'' + Cx = 0, w/ C>0 (3)
x'' - Cx = 0, w/ C>0 (4)
Ax'' + Bx' + Cx = 0, arbitrary A,B,C (5)
x'' + Cx = D sin (omega t). (6)
Here is a guide to reading (note that this includes
some sections NOT on the printed syllabus and skips
parts that are on the syllabus). Keep eqs 3-6 in mind
as you read the text.
Topic pages what to do
^^^^^ ^^^^^ ^^^^^^^^^^
The general problem statement. 129-132 (read)
Existence and Uniqueness (usually not an issue). 132-133 (skim or skip)
Linear independence of two functions. 134-135 (read quickly)
Wronskians (we will not use in this course). 135 (skip)
General solutions. 136-138 (read)
Repeated roots (a special rare case for the experts). 139 (skim or skip)
General solution of linear equations of any order. 141-149 (skip)
Homogeneous eqs with const coefficients (eqn (5) above). 152-154 (read)
Repeated roots (rare special case). 155-156 (skim or skip)
Complex variables (see also T&F A-7 to A-17). 156-157 (read, important)
Complex roots (connects sines to exponentials). 158-159 (read)
Repeated complex roots (for experts). 159-160 (skip)
Mechanical vibrations (important but a little applied). 163-170 (read if interested)
Undetermined coefficients (ie, guessing). 173-181 (read)
Variation of parameters (for experts only). 181-184 (skip)
Forcing and resonance ( eqn (6) above ). 187-195 (read lightly)
-Andy Ruina T&AM, Kimball Hall (wk) 227 Bryant Ave (hm)
ruina@cornell.edu Cornell, Ithaca,NY,14850 Ithaca, NY 14850
FAX: 607-255-2011 607-277-5675