Study Guide
Differential equations: Chapter 2
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.
Section 2.1
♠ Know the general form of a first order linear DE. Know how to
sketch a direction field for a first order linear DE in standard form.
Reference: Equation 3 on p.32
♠ Know how to find the general solution of a first order linear DE
using the integrating factor.
Reference: equations 30 and 33 on p.36
Exercise: 1-11 odd
♠ Know how to solve an IVP including a first order linear DE. Understand the relationship between solutions of IVP for first order linear DE
and direction fields.
Reference: example 4 on p.38-39. Graphs of particular solutions of the
DE are called integral curves, like in Chapter 1. These graphs are always
tangent to the slops on a direction field.
Exercise: 13-25 odd
♠♠ Turn-in problems for section 2.1: 5, 6, 14, 19
Section 2.2
♠ Know what a separable DE is.
Reference: equation 4 on p.42. Separable DE are the most common
kind of first order nonlinear DE that we can solve exactly.
♠ Know the difference between implicit and explicit solutions.
Reference: p.74. Here is the main point: an explicit solution of a DE
of the form
dy
= f (t, y)
dt
is a function written in the form y =(some function of t only). In other
words, y is given explicitly as a function of t. If the solution is not given
in this form, the solution is called implicit.
Highly suggested reading on this topic and how it relates to separable
DE: example 2 on p.45
1
2
♠ Know how to find the general solution of a separable DE. Also
know how to solve IVP involving a separable DE, and how to sketch
their integral curves.
Reference: all examples in the section
Exercise: 1-19 odd
♠♠ Turn-in problems from section 2.2: 10 part a), 14 part a), 18 part
a), 22, 20 parts a)-e), 32 parts a) and b)
Section 2.3
This section is mostly a combination of setting up DE and IVP as in
section 1.1, and solving IVP as in sections 2.1 and 2.2.
♠♠ Turn in problems from section 2.3: 17, 23
Section 2.4
The main content of this section is the existence and uniqueness theorems for first order ODE.
“Existence” and “uniqueness” are, in a way, opposite concepts. Existence means that there is at least one solution to a given IVP, and
uniqueness means that there is at most one solution (i.e., if a solution
exists, then there isn’t another one). So “existence and uniqueness” is
kind of a Goldilocks situation– the IVP is permissive enough to have a
solution (existence), but restrictive enough not to have more than one
(uniqueness). This is good for physical applications, which normally
should have one single solution to make sense!
There is a version of the existence and uniqueness theorem for linear
ODE, and another version for nonlinear ODE.
♠ Know the existence and uniqueness theorem for first order linear
ODE.
Reference: Theorem 2.4.1 on p.68. The theorem says that a unique
solution must exist on the largest interval around t0 where p(t) and g(t)
are continuous.
Exercise: 1-5
♠ Know the existence and uniqueness theorem for first order nonlinear
ODE.
Reference: Theorem 2.4.2 on p.70. The theorem says that a unique
soluion must exist on some interval around t0 as long as f (t, y) and ∂f /∂y
are continuous in a rectangle around (t0 , y0 ). Nevertheless, the solution
3
may fail to exist on the whole rectangle where f (t, y) and ∂f /∂y are
continuous.
Exercise 1: Let f (t, y) = y 2 . Find the solution u to the IVP
(1)
y 0 = f (t, y)
y(0) = 1 .
Find the interval on which u is continuous.
4
1. u(t) =
1
,
1−t
Answers to selected exercises
which is continuous on the interval (−∞, 1)