Math 315 Study Guide for Test 1 – Summer 2011
Chapter 1: Introduction to Differential Equations
This chapter is a review of basic differential and integral calculus and the ideas are used extensively
throughout the text. However, no questions will come directly from this chapter.
Chapter 2: First-Order Differential Equations
1. Equations of the form 𝑦′=𝑓(𝑡,𝑦) can be linear or nonlinear, depends on 𝑓.
What are: definition of a solution, general solution, and intervals of existence?
2. Separable equations 𝑦′=𝑓(𝑡,𝑦)=𝑔(𝑡)ℎ(𝑦) easy to solve assuming the integration is not
daunting. Understand that solutions can be explicit or implicit.
3. Models of motion – uses Newton’s laws of motion to provide a physical example of first-order
equations. You will not be asked to derive a model beyond the difficulty of say #6, p.45
4. Linear equations – main solution technique is construction of an integrating factor. The
authors introduce the terms homogeneous and particular solutions. These terms were not
emphasized during lecture. You will hear them repeatedly when we discuss Chapter 4. No
variation of parameters. We’ll see plenty of it in Chapter 4.
5. Mixing problems – you are guaranteed to see a mixi ng problem of some form. It will be one
where the integration and algebra of the problem should be straightforward.
Tip – let dimension and units guide you.
6. Exact differential equation – know how to construct the function 𝐹 where 𝑑𝐹 = 0 is equal to
the differential equation we are interested in solving. Since constructing an integrating factor
for nonlinear equations can be time consuming, they are not likely to appear on the test.
Know what homogeneous functions and homogeneous equations are and how to solve them.
The transformation 𝑦(𝑥) = 𝑣(𝑥)𝑥 transforms the equation into a separable one. Here I am
using the term homogeneous in the context of multiplicative scaling, i.e., 𝑓(𝑡𝑥,𝑡𝑦)=𝑡𝛼𝑓(𝑥,𝑦).
7. Existence and uniqueness – know Theorems 7.6 and 7.16 and understand questions such as
those on p. 86, #1 – 10 and p. 87, #25 – 32.
8. Dependence on initial conditions – nothing from this section will appear on the test.
9. Autonomous equations and stability – know what is meant by equilibrium points, stability of
equilibrium points, phase line, direction field and sketching solutions using the qualitative
methods in this section.
Chapter 3: Modeling and Applications
Material from Section 3.1 will show up on the test. Know how to use and adapt Malthusian and
logistics models, e.g., how to introduce a harvesting term into the model. Know how to write a
model in dimensionless form. This is discussed on p. 42 – 43, and I discuss the idea in this section.
Appendix: Complex Numbers and Matrices, p. 699 – 702.
The appendix is a review of some basic concepts involving complex algebra. You should know how
to add, subtract, multiply and divide complex numbers, how to find magnitude (or absolute value),
know complex conjugate, polar form of complex numbers, and the very important Euler’s formula.
Know proposition A.11.
Hopefully you tried all of the suggested problems and more!
