MECH 221 MATH LEARNING GUIDE — WEEK ONE (starts 2014-09-22)
c 2014 by Philip D. Loewen
UBC MECH 2 Materials Lecture Schedule.
2014-09-22 (Mon):
2014-09-24 (Wed):
MATH 01, Particular Solutions of L[y] = f
MATH 02, Exponential Shift, I
Learning Goals. You have mastered this week’s material when you can . . .
1. Apply a given constant-coefficient linear differential operator L to a given function y to
produce a new function L[y]. Be ready for an operator of any order n, like this:
L[y] = an y (n) + an−1 y (n−1) + . . . + a2 ÿ + a1 ẏ + a0 y.
2. Determine the characteristic polynomial associated with a given constant-coefficient linear
differential operator L. For the operator above, this is
p(s) = an sn + an−1 sn−1 + . . . + a2 s2 + a1 s + a0 .
3. Find a particular solution for any ODE of the form L[u] = q(t), with L as above and q is a
given polynomial. Know what to do in cases where (i) a0 6= 0, or (ii) a0 = 0 but a1 6= 0, or
(iii) a0 = a1 = 0 but a2 6= 0, etc.
4. Find a particular solution for any ODE of the form L[u] = ekt , where k is a real constant
that is not a root of the characteristic polynomial.
5. Apply the exponential shift formula to L to simplify L[ekt u(t)]. With L and p as above,
the formula says
(n)
p (k) (n)
p′′ (k)
p′ (k)
kt
kt
L[e u(t)] = e
u (t) + . . . +
ü(t) +
u̇(t) + p(k)u(t) .
n!
2!
1!
6. Find a particular solution for any ODE of the form L[u] = ekt , where k is a real constant
that is a root of the characteristic polynomial.
7. Use exponential shift to find a particular solution for any ODE of the form L[y] = q(t)ekt
where q(t) is a polynomial and k is a real constant. Handle cases where k is a root of the
characteristic polynomial as well as cases where it is not.
Textbook Sections. In class, we are tackling particular solutions first, before laying down all the
theory for “general solutions”. In approaching the practice problems from the textbook, focus on
finding particular solutions only. Skip over the requests for general solutions for now. (We’ll discuss
those properly in a future week.)
⊔ JL 2.5 — Nonhomogeneous equations: focus on subsection 2.5.2. Solve problems
⊓
2.5.2, 2.5.3, 2.5.5, 2.5.7(b), 2.5.9, 2.5.102(a).
⊔ WFT 5.3 — Nonhomogeneous linear equations: #1–7, 16–23, 33–38.
⊓
⊔ WFT 5.4 — The method of undetermined coefficients I: #1–14, 24–29.
⊓
Next Week’s Test. On Friday 03 October 2014, there will be a 50-minute test starting at 08:00.
The math questions will be based on topics listed on this sheet.
File “m255-2014-week01”, version of 11 Oct 2014, page 1.
Typeset at 08:52 October 11, 2014.