Test I Review Sheet--Math 308-511
General Information
The test will be given during our usual class time. It will cover the
following material: from Nagle and Saff, Chapter 1; Chapter 2, through
section 2.3; Chapter 3, through section 3.2; and Chapter 4, through
section 4.5; from Pratap (MATLAB), Chapter 1, Chapter 2, and Section
3.1. You will be allowed to use MATLAB during the test, but not MAPLE
or any program or calculator that can do symbolic manipulation. You
will be provided with a (small) table of integrals. The test will have
6 to 8 questions, some with multiple parts. One of the questions will
require you to use MATLAB. Please bring an 8½x11
bluebook.
Methods & Techniques
- First order equations
- Graphical--direction fields (Section 1.3)
- Numerical--Euler's method (Section 1.4)
- Analytical
- Separable equations (Section 2.2)
- Linear equations (Section 2.3)
- Second order equations
- Reduction of Order (Section 4.4)
- Constant coefficient and Cauchy-Euler homogeneous equations
(Section 4.5)
Foundations
- First order equations
- Existence and uniqueness, general IVP (Section 1.2)
- Forbids crossing of solutions
- Existence and uniqueness, linear IVP (Section 2.3)
- Second order linear equations
- Linear operators (Section 4.2)
- Differential equations in operator notation
- Linear combinations
- Existence and uniqueness of IVP, L[y]=g, y(x0)
= y0, y'(x0) = v0
- Homogeneous equations (Section 4.3)
- Superposition principle
- Fundamental sets
- Wronskians
- Linear independence and linear dependence
Applications
- First order equations
- Mechanics (Section 2.1, class notes)
- Newton's law of cooling (Section 2.2, web problem)
- Mixing (Sections 2.2, 2.3, 3.2)
- Population growth/decay (Section 3.2)
- Second order linear equations
- Pendulum (Section 4.1)
- Spring system (Section 5.1)
MATLAB
- Graphing
dfield5
- Plotting (Sections 2.3-2.5, Pratap)
- M-files and numerics
- Script and function M-files (Sections 2.4 & 2.5, Pratap)
- Working with arrays (Sections 2.2 & 3.1, Pratap)
- Built-in functions--
sin, cos, exp, log,...
(Section 2.1, Pratap)