Final Exam Information Sheet, Math 308-503, Fall 2014
Final dates and times
The final exam will be held on Tuesday, December 16, from 8-10 a.m. in our regular classroom.
If you have conflicts (e.g., three exams scheduled on the same day), please let me know as soon
as possible so we can make arrangements.
Exam structure
The final exam will be worth 40% of your final course grade. It will consist of roughly 10
questions, each worth about 20 points on average. You can expect each problem to be similar
in length to problems on previous exams.
“Cheat sheet”
In order to minimize the amount of memorization needed, you may bring one half sheet of paper
(8 21 × 5 12 ) filled on both sides with notes, or one full sheet of paper (8 21 × 11) filled on one side
with notes. You may type your notes if you wish. Otherwise, no aids will be allowed (calculators,
textbooks, etc.). I will provide you with a list of Laplace transforms (Table 6.2.1 in the book,
possibly modified), so you do not need to include material from this chart on your cheat sheet.
As on the first two exams, you will be expected to know integrals, derivatives, and basic properties of standard functions: polynomials and powers, exponential (ex ), logarithmic (ln x), and
basic trig (sin x, cos x, tan x). In addition, you should be able to use basic differentiation rules
(product, quotient, chain rule) and “undo” these rules in order to calculate integrals. Integration
by parts and partial fractions may be expected in the context of calculating Laplace transforms.
Material covered and expectations
The exam will cover all material for which homework was assigned, though no questions will
be explicitly drawn from sections 1.2 and 1.3. Thus the covered sections are 2.1-2.7, 3.1-3.8,
6.1-6.6, 7.1-7.6 and 7.8, and 9.1-9.3.
Questions will be similar to those on previous exams and review sheets, so these materials along
with homework assignments will serve as a good guide for studying for the final. I will also
provide a sample problem online related to the most recent material from Chapter 9. One
expectation that I would like to emphasize is that you should be able to identify (classify)
equations by type in addition to solving them once you know their type.
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Outline of material covered on the final
1. Chapter 2: First-order scalar equations
(a) Solution techniques: Identify the relevant technique and apply it!
i. Method of integrating factors (2.1)
ii. Autonomous equations (2.5): graphical techniques
iii. Separable (2.2) and exact (2.6) equations; implicit and explicit solution formulas
(b) Theory (2.4): Understand and apply Theorems 2.4.1 and 2.4.2.
(c) Modeling: mixing (2.3), finances (2.3), and population (2.3 and 2.5).
(d) Numerical methods (Euler’s method, 2.7).
2. Chapter 3: Second-order scalar equations
(a) Solution techniques: Identify the relevant technique and apply it!
i. Homogeneous equations, constant coefficients: two real roots (3.1), complex roots
(3.3), repeated roots (3.4)
ii. Nonhomogeneous equations: Undetermined coefficients (3.5) and variation of
parameters (3.6)
(b) Theory (3.2): Solutions to linear homogeneous equations and the Wronskian.
(c) Modeling: Unforced (3.7) and unforced (3.8) vibrations.
3. Chapter 6: Laplace transforms
(a) Definition of the Laplace transform (6.1): Know what the Laplace transform is, and
be able to compute simple examples using integration by parts.
(b) Step functions: Definition, and use in the representation of piecewise-defined functions.
(c) Solving initial value problems using Laplace transforms: Discontinuous forcing functions (6.4) and impulse or δ functions (6.5).
(d) The convolution integral (6.6)
4. Chapter 7: Systems of first-order linear matrics
(a) Introduction; converting higher-order equations into systems (7.1)
(b) Matrices and linear algebra: Linear independence, eigenvalues, and eigenvectors (7.2,
7.3)
(c) Basic theory of first order linear systems: Existence, uniqueness, fundamental solution
sets, and the Wronskian (7.4)
(d) Solving homogenous linear systems with constant coefficients: Three cases depending
on eigenvalue structure (7.5, 7.6, 7.8)
5. Chapter 9: Nonlinear differential equations and stability
(a) The phase plane: Type and stability of critical points in linear systems (9.1)
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(b) Autonomous systems and stability (9.2)
(c) Locally linear systems: Classifying equilibrium points of nonlinear systems by analyzing eigenvalues of the Jacobian (9.3)
Sample problem from Chapter 9
1. Consider the nonlinear system
x0 = (2 + x)(y − x),
y 0 = y(2 + x − x2 ).
a) Determine the equilibrium points of the above system.
b) Find the corresponding linear system near each critical point.
c) Find the eigenvalues of each linear system.
d) Specify what information about each critical point (type and stability) can be drawn from
the eigenvalues found in part c).
Note: You will not be asked to construct a phase portrait on the exam. You may however be
asked questions concerning phase portraits given on the exam, for example by matching a given
nonlinear system with its phase portrait in a multiple choice problem.
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Solutions:
1. Consider the nonlinear system
x0 = (2 + x)(y − x),
y 0 = y(2 + x − x2 ).
a) Determine the equilibrium points of the above system. Solution: Factor
yields four
analysis
−2
sets of equations. First, 2 + x = 0 and y = 0 yields the critical point
. The second is
0
0
y = x and y = 0, yielding
. Thirdly, 2 + x = 0 and 2 + x − x2 = 0 has no solution. Finally,
0
2
−1
y − x = 0 and 2 + x − x2 = 0 yields
and
.
2
−1
b)
The Jacobian is
Find the corresponding linear system near each critical point. Solution:
2
0
−2
y − 2 − 2x
2+x
~u.
is ~u0 =
, so the corresponding linear system at
0 −4
0
y(1 − 2x) 2 + x − x2
−1
−4 4
2
−2 2
0
~u. Finally, at
we have ~u0 =
~u. At
we have ~u0 =
At
−1
−6 0
2
0 2
0
−1 1
we have ~u0 =
~u.
−3 0
c,d) Find the eigenvalues of each linear system. Solution. (Eigenvalue
are omitted;
computations
2
0
are 2 and -4, so
you would have to show these on the exam!) The eigenvalues of
0 −4
−2
is an unstable saddle. The eigenvalues
for the nonlinear system the equilibrium point
0
−4 4
0
−2 2
are
is an unstable saddle. The eigenvalues of
are -2 and 2, so
of
−6 0
0
0 2
√
−1 1
2
−2 ± i2 5, so
is an asymptotically stable spiral. Finally, the eigenvalues of
−3 0
2
√
−1
are − 12 ± i 211 , so
is also an asymptotically stable spiral.
−1
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