Study Guide
Introduction to differential equations
How to use this study guide:
You should understand each ♠ point, by using the suggested resources
or any others. Corresponding exercises are recommended, but not required. Answers for exercises not in the book are given at the end of this
guide.
Each ♠♠ denotes required turn-in homework.
♠ Understand how to write down a differential equation (DE) from a
word problem using Newton’s 2nd law.
Reference: Ex. 1.1.1 on p.1-3
Exercise: 1.1.25
♠ Understand how to write down a DE from a word problem in other
situations, including population size problems, chemical concentration
problems, flow problems, etc.
Writing down a DE from a word problem is an important skill for this
course. I recommend following these steps:
•
•
•
•
Read the problem
Define the independent variable and its units
Define the dependent variable and its units
Identify where the DE is going to come from (e.g. Newton’s 2nd
law, fluids flowing in and out of a container, ...)
• Write down the DE. Usually the DE has the form du
= (...some
dt
terms involving u and t...)
• Verify that all terms in the DE have the same units.
Reference: “Field mice and owls” on bottom of p.5
Exercise: 1.1.22
♠ What is a direction field? How is one drawn for a DE?
Reference: Example 2 on p.3-4
Exercise: 1.1.15-1.1.20
♠ What is an equilibrium solution? How are they found? How can
they be seen on a direction field?
Reference: Example 2 on p.3-4
♠♠ Homework turn-in problems from section 1.1: 21, 23
1
2
♠ Know that a “solution of a DE” is a function that solves the equation.
Exercise 1: Suppose u(t) is a solution of the differential equation
dy
= 5y − 3t
dt
and that u(4) = 2.
a) find u0 (4)
b) u00 (4).
♠ What is an initial condition (IC)? What is an initial value problem
(IVP)? What does it mean for a function to solve an IVP?
Reference: p.12
♠ Understand how to solve certain simple differential equations.
Reference: Example 1.2.1 on p.10-11
Exercise: 1.2.11
♠♠ Homework turn in problems from section 1.2: 7, 18
♠ What is an ordinary differential equation (ODE)? What is a partial
differential equation (PDE)?
Reference: p.19
♠ What is a system of DE?
Reference: Bottom of p.19 to p.20
♠ What is the order of a DE?
Reference: p.20
♠ What is a linear DE? Understand how to tell if a given DE is linear.
Reference: Formula (11) on p.21
Although formula (11) is the formal definition of a linear DE, it is often
useful to use the following telltale sign of a nonlinear DE: if there is a term
in which the dependent variable y or one of its derivatives appears raised
to any power, has some function acting on it besides being multipled by
a constant or a function of t, or y or any of its derivatives are multiplied
together, then the DE is nonlinear.
♠♠ Homework turn-in problems from section 1.3: 5, 8, 17, 19, 30
3
Answers to selected exercises
1. a) -2, b) -13