Fundamentals of differential equations

697 Pages·2014·6.98 MB·English

by  Nagle, R. Kent, Saff, E. B., Snider, Arthur David

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ISBN 978-1-29202-382-3 8
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9 781292 023823
Fundamentals of Differential Equations
Nagle Saff Snider
Eighth Edition
ISBN 10: 1-292-02382-1
ISBN 13: 978-1-292-02382-3
Pearson Education Limited
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Essex CM20 2JE
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ISBN 10: 1-292-02382-1
ISBN 13: 978-1-292-02382-3
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Printed in the United States of America
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P E A R S O N C U S T O M L I B R AR Y
Table of Contents
1. Introduction
R. Kent Nagle/Edward B. Saff/Arthur David Snider 1
2. First-Order Differential Equations
R. Kent Nagle/Edward B. Saff/Arthur David Snider 37
3. Mathematical Models and Numerical Methods Involving First-Order Equations
R. Kent Nagle/Edward B. Saff/Arthur David Snider 93
4. Linear Second-Order Equations
R. Kent Nagle/Edward B. Saff/Arthur David Snider 159
5. Introduction to Systems and Phase Plane Analysis
R. Kent Nagle/Edward B. Saff/Arthur David Snider 253
6. Theory of Higher-Order Linear Differential Equations
R. Kent Nagle/Edward B. Saff/Arthur David Snider 335
7. Laplace Transforms
R. Kent Nagle/Edward B. Saff/Arthur David Snider 369
8. Series Solutions of Differential Equations
R. Kent Nagle/Edward B. Saff/Arthur David Snider 445
9. Matrix Methods for Linear Systems
R. Kent Nagle/Edward B. Saff/Arthur David Snider 525
10. Partial Differential Equations
R. Kent Nagle/Edward B. Saff/Arthur David Snider 599
Tables and Equations
R. Kent Nagle/Edward B. Saff/Arthur David Snider 683
Index 687
I
II
Introduction
1
BACKGROUND
In the sciences and engineering,mathematical models are developed to aid in the understanding
of physical phenomena. These models often yield an equation that contains some derivatives
of an unknown function. Such an equation is called a differential equation. Two examples of
models developed in calculus are the free fall of a body and the decay of a radioactive substance.
In the case of free fall, an object is released from a certain height above the ground and
falls under the force of gravity.†Newton’s second law,which states that an object’s mass times
its acceleration equals the total force acting on it, can be applied to the falling object. This
leads to the equation (see Figure 1)
d2h
m (cid:3)(cid:2)mg ,
dt2
where mis the mass of the object,his the height above the ground,d2h/dt2 is its acceleration,
gis the (constant) gravitational acceleration,and (cid:2)mgis the force due to gravity. This is a differ-
ential equation containing the second derivative of the unknown height has a function of time.
Fortunately,the above equation is easy to solve for h. All we have to do is divide by mand
integrate twice with respect to t. That is,
d2h
(cid:3)(cid:2)g ,
dt2
so
dh
(cid:3)(cid:2)gt(cid:4)c
dt 1
and
(cid:2)gt2
h(cid:3)h t (cid:3) (cid:4) c t (cid:4)c .
A B 2 1 2
−mg
h
Figure 1Apple in free fall
†We are assuming here that gravity is the onlyforce acting on the object and that this force is constant. More general
models would take into account other forces,such as air resistance.
From Chapter 1 of Fundamentals of Differential Equations,Eighth Edition. R. Kent Nagle,Edward B. Saff and
Arthur David Snider. Copyright ©2012 by Pearson Education,Inc. Publishing as Pearson Addison-Wesley. All rights reserved.
1
Introduction
We will see that the constants of integration, c and c , are determined if we know the initial
1 2
height and the initial velocity of the object. We then have a formula for the height of the object
at time t.
In the case of radioactive decay (Figure 2), we begin from the premise that the rate of
decay is proportional to the amount of radioactive substance present. This leads to the equation
dA
(cid:3)(cid:2)kA , k(cid:5)0 ,
dt
where A (cid:5)0 is the unknown amount of radioactive substance present at time tand kis the pro-
A B
portionality constant. To solve this differential equation,we rewrite it in the form
1
dA(cid:3)(cid:2)k dt
A
and integrate to obtain
(cid:2) 1 (cid:2)
dA(cid:3) (cid:2)k dt
A

ln A(cid:4)C (cid:3)(cid:2)kt(cid:4)C .
1 2
Solving for Ayields
A(cid:3)A t (cid:3)eln A(cid:3)e(cid:2)kteC2(cid:2)C1 (cid:3)Ce(cid:2)kt ,
A B
where C is the combination of integration constants eC2(cid:2)C1. The value of C, as we will see
later,is determined if the initial amount of radioactive substance is given. We then have a for-
mula for the amount of radioactive substance at any future time t.
Even though the above examples were easily solved by methods learned in calculus,they do
give us some insight into the study of differential equations in general. First,notice that the solu-
tion of a differential equation is a function,like h t or A t ,not merely a number. Second,inte-
A B A B
gration is an important tool in solving differential equations (not surprisingly!). Third,we cannot
expect to get a unique solution to a differential equation,since there will be arbitrary “constants
of integration.”The second derivative d2h/dt2in the free-fall equation gave rise to two constants,
c and c ,and the first derivative in the decay equation gave rise,ultimately,to one constant,C.
1 2
Whenever a mathematical model involves the rate of changeof one variable with respect
to another, a differential equation is apt to appear. Unfortunately, in contrast to the examples
for free fall and radioactive decay,the differential equation may be very complicated and diffi-
cult to analyze.
A
Figure 2Radioactive decay
2
Introduction
R L
+
emf C
−
Figure 3Schematic for a series RLC circuit
Differential equations arise in a variety of subject areas,including not only the physical sci-
ences but also such diverse fields as economics,medicine,psychology,and operations research.
We now list a few specific examples.
1. In banking practice,if P t is the number of dollars in a savings bank account that
A B
pays a yearly interest rate of r% compounded continuously,then Psatisfies the dif-
ferential equation
dP r
(1) (cid:3) P , t in years.
dt 100
2. A classic application of differential equations is found in the study of an electric cir-
cuit consisting of a resistor,an inductor,and a capacitor driven by an electromotive
force (see Figure 3). Here an application of Kirchhoff’s laws leads to the equation
d2q dq 1
(2) L (cid:4) R (cid:4) q(cid:3)E t ,
dt2 dt C A B
where Lis the inductance,Ris the resistance,Cis the capacitance,E t is the electro-
A B
motive force,q t is the charge on the capacitor,and tis the time.
A B
3. In psychology,one model of the learning of a task involves the equation
dy/dt 2p
(3) (cid:3) .
y3/2 1(cid:2)y 3/2 1n
A B
Here the variable yrepresents the learner’s skill level as a function of time t. The
constants pand ndepend on the individual learner and the nature of the task.
4. In the study of vibrating strings and the propagation of waves,we find the partial
differential equation
02u 02u
(4) (cid:2)c2 (cid:3)0 ,††
0t2 0x2
where trepresents time,xthe location along the string,cthe wave speed,and uthe
displacement of the string,which is a function of time and location.
††Historical Footnote:This partial differential equation was first discovered by Jean le Rond d’Alembert (1717–1783)
in 1747.
3
Introduction
To begin our study of differential equations, we need some common terminology. If an
equation involves the derivative of one variable with respect to another, then the former is
called a dependent variable and the latter an independent variable. Thus,in the equation
d2x dx
(5) (cid:4)a (cid:4)kx(cid:3)0 ,
dt2 dt
tis the independent variable and xis the dependent variable. We refer to aand kas coefficients
in equation (5). In the equation
0u 0u
(6) (cid:2) (cid:3)x(cid:2)2y ,
0x 0y
xand yare independent variables and uis the dependent variable.
A differential equation involving only ordinary derivatives with respect to a single indepen-
dent variable is called an ordinary differential equation. A differential equation involving partial
derivatives with respect to more than one independent variable is a partial differential equation.
Equation (5) is an ordinary differential equation,and equation (6) is a partial differential equation.
The orderof a differential equation is the order of the highest-order derivatives present in
the equation. Equation (5) is a second-order equation because d2x/dt2 is the highest-order
derivative present. Equation (6) is a first-order equation because only first-order partial deriva-
tives occur.
It will be useful to classify ordinary differential equations as being either linear or nonlin-
ear. Remember that lines (in two dimensions) and planes (in three dimensions) are especially
easy to visualize,when compared to nonlinear objects such as cubic curves or quadric surfaces.
For example, all the points on a line can be found if we know just two of them. Correspond-
ingly,lineardifferential equations are more amenable to solution than nonlinear ones. Now the
equations for lines ax(cid:4)by(cid:3)cand planes ax(cid:4)by(cid:4)cz(cid:3)dhave the feature that the variables
appear in additive combinations of their first powers only. By analogy a linear differential
equationis one in which the dependent variable yand its derivatives appear in additive combi-
nations of their first powers.
More precisely,a differential equation is linearif it has the format
dny dn(cid:3)1y p dy
(7) anAxB dxn (cid:2) an(cid:2)1AxB dxn(cid:3)1 (cid:2) (cid:2)a1AxB dx (cid:2)a0AxBy(cid:4) FAxB ,
where anAxB,an(cid:2)1AxB,. . . ,a0AxB and FAxB depend only on the independent variable x. The addi-
tive combinations are permitted to have multipliers (coefficients) that depend on x; no restric-
tions are made on the nature of this x-dependence. If an ordinary differential equation is not
linear,then we call it nonlinear. For example,
d2y
(cid:4) y3(cid:3)0
dx2
is a nonlinear second-order ordinary differential equation because of the y3term,whereas
dx
t3 (cid:3)t3(cid:4) x
dt
is linear (despite the t3terms). The equation
d2y dy
(cid:2)y (cid:3)cos x
dx2 dx
is nonlinear because of the y dy/dx term.
4
Introduction
Although the majority of equations one is likely to encounter in practice fall into the nonlin-
ear category,knowing how to deal with the simpler linear equations is an important first step (just
as tangent lines help our understanding of complicated curves by providing local approximations).
1
EXERCISES
In Problems 1–12,a differential equation is given along dp
8. (cid:3)kp P(cid:2)p ,where kand Pare constants
with the field or problem area in which it arises. Classify dt A B
each as an ordinary differential equation (ODE) or a (logistic curve,epidemiology,economics)
partial differential equation (PDE), give the order, and
d4y
indicate the independent and dependent variables. If the 9. 8 (cid:3)x 1(cid:2)x
dx4 A B
equation is an ordinary differential equation, indicate
(deflection of beams)
whether the equation is linear or nonlinear.
d2y dy
d2y dy 10. x (cid:4) (cid:4)xy(cid:3)0
1. (cid:2) 2x (cid:4) 2y (cid:3) 0 dx2 dx
dx2 dx
(aerodynamics,stress analysis)
(Hermite’s equation,quantum-mechanical harmonic 0N 02N 1 0N
11. (cid:3) (cid:4) (cid:4)kN, where kis a constant
oscillator) 0t 0r2 r 0r
d2x dx (nuclear fission)
2. 5 (cid:4) 4 (cid:4) 9x(cid:3)2 cos 3t
dt2 dt d2y dy
12. (cid:2)0.1 1(cid:2)y2 (cid:4)9y(cid:3)0
(mechanical vibrations,electrical circuits,seismology) dx2 A B dx
(van der Pol’s equation,triode vacuum tube)
02u 02u
3. (cid:4) (cid:3)0
0x2 0y2 In Problems 13–16,write a differential equation that fits
the physical description.
(Laplace’s equation,potential theory,electricity,heat,
aerodynamics) 13. The rate of change of the population pof bacteria at
time tis proportional to the population at time t.
dy y 2(cid:2)3x
4. (cid:3) A B 14. The velocity at time t of a particle moving along a
dx x 1(cid:2)3y straight line is proportional to the fourth power of its
A B
position x.
(competition between two species,ecology)
15. The rate of change in the temperature Tof coffee at
dx
5. (cid:3)k 4(cid:2)x 1(cid:2)x ,where kis a constant time t is proportional to the difference between the
dt A BA B
temperature M of the air at time t and the tempera-
(chemical reaction rates) ture of the coffee at time t.
dy 2 16. The rate of change of the mass Aof salt at time tis
6. y 1(cid:4) (cid:3)C,where Cis a constant proportional to the square of the mass of salt present
c adxb d
at time t.
(brachistochrone problem,†calculus of variations)
17. Drag Race. Two drivers,Alison and Kevin,are par-
d2y dy ticipating in a drag race. Beginning from a standing
7. 11 (cid:2) y dx2 (cid:4) 2x dx (cid:3) 0 start,they each proceed with a constant acceleration.
Alison covers the last 1/4 of the distance in 3 sec-
(Kidder’s equation, flow of gases through a porous onds, whereas Kevin covers the last 1/3 of the dis-
medium) tance in 4 seconds. Who wins and by how much time?
†Historical Footnote:In 1630 Galileo formulated the brachistochrone problem bra´x´isto(cid:6)(cid:3)shortest,xro´no(cid:6)(cid:3)time,that is,to determine a
A B
path down which a particle will fall from one given point to another in the shortest time. It was reproposed by John Bernoulli in 1696 and solved by
him the following year.
5
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