Test 1 Review — Math 308502 (Summer
2015)
General Information
Test 1 will be given on Friday, 7/3/15, during our usual class
time and in our usual classroom. I will have extra office hours on
Thursday afternoon, 14 pm, and on Friday morning, 9:3010:30 am.
 Bluebooks. Please bring an 8½x11 bluebook.

Calculators. You may use scientific calculators to do numerical
calculations — logs, exponentials, and so on. You
may not use any calculator that has the capability of doing
algebra or calculus, or of storing course material.

Other devices. You may not use cell phones, computers, or any
other device capable of storing, sending, or receiving information.

Structure and Coverage There will be 6 to 8 questions, some
with multiple parts. The problems will be similar to homework
problems, examples done in class and worked out in the text. The
material covered is from the following sections in the text:
1.2, 2.12.6, 3.13.7.
Topics covered
First Order ODEs
 Methods
 Integrating factors— Be able to find and use integrating
factors to solve 1st order linear ODEs, including initial value
problems.
 Separable equations— Be able to determine when an equation
is separable and know how to solve the equation, and able to
solve initial value problems.
 Nonlinear vs. Linear equations— Elementary
requirements for most physical models: A solution must exist
and there must be only one solution. Linear equations have these
properties; however, nonlinear ones may not.
 Exact equations — Be able to determine whether or not
an equation is exact. Be able to solve exact equations, including
initial value problems.
 Applied Problems
 Mixing problems — Set these up using dx/dt =
rate in − rate out. Be sure to check units!

Newton's law of cooling — Temperature in a building and
"timeofdeath" problems
 Circuits — Be able to set up and solve simple
circuits.

Falling bodies & rockets — Air friction is modeled by
friction force = −bv. Be able to find limiting velocities
(section 1.2). Be able to find the escape velocity of a rocket.
 Population dynamics. — Logistic model. Gompertz
model (problem 2.5.16). Phase line. Equilibrium
points. Stability of an equilibrium point.
Second Order ODEs
 Foundations
 Linear equations, initial value problems, operator notation
 Homogeneous linear equations: principle of superposition,
Wronskians
 Wronskians, fundamental sets, Abel's theorem. (Be able to
state and prove this.)
 Nonhomogeneous equations: Any two solutions differ by a
homogeneous solution
 Superposition of forces
 Methods
 Constant coefficient homogeneous equations: Find
characteristic polynomial. Three cases: two distinct real roots; one
double real root; complex roots. Be able to solve initial value
problems. (Sections 3.1, 3.3)
 Reduction of order. Use this to find a second solution for the
fundamental set. (Section 3.2)
 Undetermined coefficients. The table on p. 181 will be given.
 Variation of parameters. Formula (26), p. 188 will be
given.
 Applied problems.
 Be able to solve initial value problems.
 Be able to set up and solve free vibration problems  springs
and circuits. Terms you should know and be able to find: undamped,
natural frequency and period, damped frequency and period, amplitude
and phase. (Section 3.7)
Updated 6/29/2015.