in which x(t) is the position at time t of an object of mass m suspended on a spring,
and c and k are the damping coecient and spring constant, respectively. The function f
represents an external forc
in volts. Suppose that both the charge Q and the current I are zero initially.
(a) Show that the differential equation governing the charge Q(t) is
Q00 + 12Q0 + 36Q = 24 cos 6t
(b) Find the charge Q(t
since we will be considering only ordinary dierential equations, the dependent variable is a
function of a single independent variable. In addition to the independent and dependent
variables, a third
which can be solved explicitly for x by applying the exponential function:
eln |x+1| = |x + 1| = e(t
2
/2+C)
= eC e(t
2
/2)
.
If the positive constant eC is allowed to take on both positive and negati
22. (Population growth) Repeat problem 21, but this time assume the growth law dP
dt =
rP (1 P/300) (this is equation (2.6) with carrying capacity N = 300). The solution
to the dierential equation is
Exercises 1.2 In each of the problems 1 4, use dierentiation to show that the given
function is a solution of the equation for all values of the constants (and hence is a general
solution):
1. Functio
Chapter 3. Second-order Dierential Equations
3.1 General Theory of Second-order Linear Dierential Equations
3.2 Homogeneous Constant Coecient Linear Dierential Equations
3.2.1 Initial Conditions
3.2.2
1.2.4
Finite time blow-up
For some initial value problems questions about long-term behavior dont make sense, because the solution runs into a vertical asymptote, and hence goes to in a nite amount of
Example 1.1.4 Write the second-order dierential equation y + 2y + 2y = 0 as a system
of two rst-order dierential equations.
We dene y1 = y, and y2 = y . Now solve for y in the dierential equation to g
2.2
Geometric Method, the Slope Field
For any rst-order dierential equation
x = f (t, x),
(2.10)
whether or not it can be solved by some analytic method, it is possible to obtain a large
amount of gra
1.2
1.2.1
General Solutions and Initial Value Problems
General solutions
In the last section in Example 1.1.1 we saw that there can be more than one solution to a
dierential equation. In the next exam
(a) y 0 + y = 3e2t has a solution represented by graph:
(b) y 00 y = 0 has a solution represented by graph:
(c) y 00 + y = 0 has a solution represented by graph:
(d) 16y 00 8y 0 + 17y = 16 cos 2t 47 s
A well-known system of two rst-order dierential equations with two dependent variables
is the Lotka-Volterra system of equations (also known as the Predator-Prey equations)
dx
= ax bxy
dt
dy
= cy + dx
The logistic growth equation (2.7) is a separable dierential equation, but the expression
dP/[P (1 P/N )] has to be integrated using partial fractions, or with the use of computer
algebra. In either c
A simple device can be employed to make solving separable equations a bit more straightforward, and it also avoids the integration by substitution. If, in equation (2.3), we split the
dierential dx/dt
vector at that point. Note that in order for this to work, it has to be assumed that the
direction of the slope vectors changes continuously in both the t and x directions. It appears
that all solutio
1.2.6
Dierent Ways of Solving Dierential Equations
One of the most intriguing things about the study of dierential equations, from the point
of view of a mathematician, is that for the majority of die
6.4.2 Transfer Functions and the Frequency Domain
6.4.3 Filters and the Response Curve
6.4.4 Poles and Zeros
Appendix A: Answers to Odd-numbered Exercises
Appendix B: Derivative and Integral Formulas
Chapter 2
First-order Dierential
Equations
In this chapter, methods will be given for solving rst-order dierential equations. Remember that rst-order implies that the rst derivative of the unknown fun
0.1
Preface
This text is intended for a one semester course in dierential equations, as is normally taken
by engineering, science, and mathematics majors in their sophmore year of college. The
prerequ
Exercises 2.1 Determine whether or not each equation is separable.
1. x + 2x = et
2. x + 2x = 1
3. x =
x+1
t+1
4. x =
sin t
cos x
Put each equation below into the form x (t) = g(t)h(x), and solve it b
A T (t) = ekt ,
where is eC . The explicit solution is
T (t) = A ekt .
(2.9)
The long term behavior is very easy to determine here, since T A as t . Thus the
temperature of the small body tends to the
Chapter 1
Introduction to Dierential
Equations
Dierential equations arise from real-world problems and problems in applied mathematics.
One of the rst things you are taught in Calculus is that the der
14. x = (t + 2)/x, x(t) =
t2 + 4t + 1
In the next two problems below, show that the given functions form a solution. Determine the largest interval of the independent variable over which the solution
Table of Contents
Preface
Chapter 1. Introduction to Dierential Equations
1.1 Basic Terminology
1.2 General Solutions and Initial Value Problems
1.2.1 General Solutions
1.2.2 Initial Value Problems
1.
1.2.2
Initial value problems
Sometimes, instead of the entire family of solution curves corresponding to a dierential
equation, we are interested in one particular curve which goes through a specic po
p (t) = et .
With the three functions p(t), p (t) and p (t) substituted into the dierential equation in
place of x, x , and x , it becomes
(et ) + 3(et ) + 2(et ) = (1 3 + 2)(et ) = (0)(et ) 0,
which
that is, if the function f (t, x) can be factored into a product of a function of t times a
function of x. Such a dierential equation is called separable.
Example 2.1.1 Determine which of the followin