Test 1 Review Math 308-503 (Spring
2011)
General Information
Test 1 will be given on Wednesday, 2/16/09, during our usual class
time and in our usual classroom. I will have extra office hours on
Tuesday afternoon, 1-4 pm, and on Wednesday morning, 8:30-9:30 am.
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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.
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Other devices. You may not use cell phones, computers, or any
other device capable of storing, sending, or receiving information.
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Structure and Coverage There will be 4 to 6 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: 1.1, 1.2,
2.1-2.4, 2.6 (exact equations only), 3.1-3.3.
Topics covered
Applications of First Order ODEs
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Falling bodies & rockets Air friction is modeled by
friction force = −bv. Be able to find limiting
velocities. (1.1, examples 1, 2 and problem 25; 1.2, example 2 and
problem 15; 2.3, example 4 and problem 29)
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Newton's law of cooling Temperature in a building and
"time-of-death" type problems (1.1, example 4 and problem 23; 1.2,
example 2 and problem 15; 2.3, problems 16, 18; in-class example,
1/31)
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Mixing problems Set these up using amounts, applying
dx/dt = rate in − rate out. If the problem requires
concentration, get it from this equation: concentration =
amount/volume. (1.1, problems 21, 24; 1.2, problems 22, 23; 2.3,
examples 1, 3 and problems 4, 5; 3.2, example 1 and problem 24; 3.3,
example 1)
- Circuits Be able to set up and solve simple
circuits. (1.1, problem 21; 3.2, problems 21, 22, 23; 3.3, problem
27; in-class example, 2/9 and 2/11)
Solving First Order ODEs
- Integrating factors Be able to find and use integrating
factors to solve 1st oder linear ODEs, including initial value
problems (2.1, examples 1-3 and problems 7, 8, 16-19)
- Separable equations Be able to determine when an equation
is sepaprable and know how to solve the equation. Also, be able to
solve initial value problems (2.2)
- 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. (2.4, examples 3,4;
in-class notes, 2/2)
- Exact equations Be able to determine whether or not
an equation is exact. Be able to solve exact equations, including
initial value problems (2.6, examples 1-3, problems 13, 16)
Systems of Two First Order Linear ODEs
- 2×2 matrices Know how to do basic matrix
operations: addition, mutliplication, inversion (magic formula!),
determinats, solving systems of two equations in two unknowns. Know
how to solve eigenvalue problems. (3.1, examples 1-7 and assigned
problems)
- Solving homogeneous systems & intial value problems Be
able to solve homogeneous equations, X′ = AX, plus initial
value problem X(0) = X0. The method is the following:
- Solve the eigenvalue problem Av = λv to find the
eigenvalues λ1, λ2 and
corresponding eigenvectors v1, v2.
- Form the general solution, X(t) = c1
eλ1t v1 + c2
eλ2t v2. (If only the general
solution is required, then stop here.)
- To solve the initial value problem, you need to get the two
coefficients c1 and c1. Set t = 0 in the
general solution to get X(0) = X0 =
c1v1 + c2v2. This is
best done by viewing the system is matrix form
[v1 v2]C = X0
where C = column vector with entries c1 and c2
See section 3.3, examples 1-4, assignmed problems; in-class notes,
2/11.
Updated 2/13/2011.