Table Of ContentSEVENTH EDITION
FU N DAM E N TALS O F
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SEVENTH
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EDITION
ISBN-13: 978-0-321-97710-6
ISBN-10: 0-321-97710-6
9 0 0 0 0
NAGLE
SAFF
SNIDER
9 780321 977106
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A BRIEF TABLE OF INTEGRALS* (continued)
1u sin u du = sin u - u cos u. 1un sin u du = -un cos u + n1un-1 cos u du.
1u cos u du = cos u + u sin u. 1un cos u du = un sin u - n1un-1 sin u du.
eau a sin nu - n cos nu eau a cos nu + n sin nu
1eau sin nu du = . 1eau cos nu du = .
a2 + n2 a2 + n2
1 2 1 2
sin a + b u sin a - b u
1sin au sin bu du = - + , a2 ≠ b2.
2 a + b 2 a - b
1 2 1 2
sin 1a + b 2u sin 1a - b 2u
1cos au cos bu du = + , a2 ≠ b2.
2 a + b 2 a - b
1 2 1 2
c1os a +2b u c1os a -2b u
1sin au cos bu du = - - , a2 ≠ b2.
2 a + b 2 a - b
1 2 1 2
1sinh u du = cosh u. 1 2 1 2 1cosh u du = sinh u.
∞
Γ t = e-uut-1du, t 7 0; Γ 1 = p; and Γ n + 1 = n!, if n is a positive integer.
2
L
0 2
1 2 1 2 1 2
SOME POWER SERIES EXPANSIONS
f ″ a f n a
f x = f a + f′ a x - a + 2! x - a 2 + g + 1 n2! x - a n + g Taylor series
1 2 1 2
1 2 1 2 1 21 2 1 2 1 2 1 2
∞ xn ∞ -1 nx2n+1 ∞ -1 nx2n
ex = a sin x = a cos x = a
n=0 n! n=01 2n2+ 1 ! n=01 2n2 !
∞ 1 ∞ 2 1 ∞2xn
1 - x -1 = axn 1 - x -2 = a n + 1 xn ln 1 - x = - a
n
n=0 n=0 n=1
1 2 1 2 1 2 1 2
tan x = x + 13 x3 + 125 x5 + 31175 x7 + 268235 x9 + g ∞ x2n+1
arcsin x = x + x#3 + 1# #3# x5 + 1# #3# #5# x7 + g arctan x = na=0 -1 n2n + 1
2 3 2 4 5 2 4 6 7 1 2
∞ -1 kx2k ∞ -1 kx2k+1 ∞ -1 kx2k+n
J0 x = ka=0 1k! 2222k J1 x = ka=0 k!1k +21 !22k+1 Jn x = ka=0 k!Γ 1n + k2+ 1 22k+n
1 2 1 2 1 2
1 2 1 2 1 2
*Note: An arbitrary constant is to be added to each indefinite integral.
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LINEAR FIRST-ORDER EQUATIONS
The solution to y′ + P x y = Q x taking the value y x0 at x = x0 is
x
y x = e-Lxx0 P j1d2j eL1xj0 2P z dz Q j dj + y 1x0 2 .
L
1 2 x0 1 2
1 2 c 1 2 1 2d
METHOD OF UNDETERMINED COEFFICIENTS
To find a particular solution to the constant-coefficient differential equation
ay″ + by′ + cy = Pm t ert ,
where P t is a polynomial of degree m, use the form
m 1 2
yp 1t 2= ts Amtm + g + A1t + A0 ert ;
if r is no1t a2 root o1f the associated auxiliary2 equation, take s = 0; if r is a simple root of
the associated auxiliary equation, take s = 1; and if r is a double root of the associated
auxiliary equation, take s = 2.
To find a particular solution to the differential equation
ay″ + by′ + cy = Pm t eat cos bt + Qn t eat sin bt , b ≠ 0 ,
where P t is a polynomial of degree m and Q t is a polynomial of degree n, use the
m 1 2 1 2n
form
1 2 1 2
yp t = ts Aktk + g + A1t + A0 eat cos bt
1 2 +1ts Bktk + g + B1t +2B0 eat sin bt ,
where k is the larger1 of m and n. If a + ib is2 not a root of the associated auxiliary equa-
tion, take s = 0; if a + ib is a root of the associated auxiliary equation, take s = 1.
VARIATION OF PARAMETERS FORMULA
If y1 and y2 are two linearly independent solutions to ay″ + by′ + cy = 0, then a
particular solution to ay″ + by′ + cy = f is y = y1y1 + y2y2, where
-f t y2 t f t y1 t
y1 t = LaW 1y12, y21 2t dt, y2 t = LaW1y21, y12 2t dt,
1 2 1 2
and W y1, y2 t = y1 t3y2= t 4-1 2y1= t y2 t . 3 41 2
3 41 2 1 2 1 2 1 2 1 2
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A TABLE OF LAPLACE TRANSFORMS
f t F s = ℒ f s f t F s = ℒ f s
1 2 1 2 5 6 1 2 1 2 1 2 5 6 1 2
1 s 1 p
1. f at F 20.
a a t 2s
1 2 a b 2 2
p
2. eat f t F s-a 21. t
22s3/2
2
1 2 1 2 # #
3. f′ t sF s -f 0 22. tn- 1 2 , n = 1, 2,c 1 3 5g 2n-1 p
2nsn+ 1/2 2
1 > 2 1 2
1 2 1 2 1 2 Γ r+1 1 2
4. f n t snF s -sn-1f 0 -sn-2f′ 0 23. tr, r7 -1
sr+1
1 2 1 2
1 2 1 2 1 2 1 2 b
- g- sf n-2 0 -f n-1 0 24. sin bt s2+b2
1 2 1 2
5. tn f t -1 nF n s 1 2 1 2 25. cos bt s
s2+b2
1 2
1 1 2 1 2 1 2 b
6. t f t 1s∞F u du 26. eat sin bt s-a 2+b2
7. 10t 1 f 2y dy Fss 1 2 27. eat cos bt 1s-sa-22a+b2
1 2
1 2 b
8. f * g t F s G s 28. sinh bt 1 2
s2-b2
1 21 2 110Te2-stf1t2dt s
9. f t+T = f t 29. cosh bt
1-e1-sT2 s2-b2
1 2 1 2 2b3
10. f t-a u t-a , aÚ0 e-asF s 30. sin bt-bt cos bt
s2+b2 2
1 2 1 2 1 2 2bs
11. g t u t-a , aÚ0 e-as ℒ g t+a s 31. t sin bt 1 2
s2+b2 2
1 2 1 2 e-as 5 1 261 2 2bs2
12. u t-a , aÚ0 32. sin bt+bt cos bt 1 2
s s2+b2 2
13. q1 a,b t2, 06a6b e-sa-s e-sb 33. t cos bt 1ss22+-bb2222
1 2 4b3
14. d t-a , aÚ0 e-as 34. sin bt cosh bt- cos bt sinh bt 1 2
s4+4b4
1 2 1 2b2s
15. eat 35. sin bt sinh bt
s-a s4+4b4
n! 2b3
16. tn, n = 1, 2,c 36. sinh bt- sin bt
sn+1 s4-b4
n! 2b2s
17. eattn, n = 1, 2,c 37. cosh bt- cos bt
s-a n+1 s4-b4
18. eat-ebt 1s-1aa2-sb2-b 38. Jv bt , v7 -1 12bvs2+s2b+2-b2s2v
a-b s 1 2 2
19. aeat-bebt 1 21 2
s-a s-b
1 2
1 21 2
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SEVENTH EDITION
Fundamentals of
Differential Equations
and Boundary Value
Problems
R. Kent Nagle
Edward B. Saff
Vanderbilt University
Arthur David Snider
University of South Florida
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Library of Congress Cataloging-in-Publication Data
Names: Nagle, R. Kent. | Saff, E. B., 1944- | Snider, Arthur David, 1940-
Title: Fundamentals of differential equations and boundary value problems.
Description: Seventh edition / R. Kent Nagle, Edward B. Saff, Vanderbilt
University, Arthur David Snider, University of South Florida. | Boston :
Pearson, [2018] | Includes index.
Identifiers: LCCN 2016030692| ISBN 9780321977106 (hardcover) | ISBN
0321977106 (hardcover)
Subjects: LCSH: Differential equations—Textbooks. | Boundary value
problems—Textbooks.
Classification: LCC QA371 .N243 2018 | DDC 515/.35—dc23
LC record available at https://lccn.loc.gov/2016030692
1—16
Student Edition ISBN13: 978-0-321-97710-6
www.pearsonhighered.com Student Edition ISBN10: 0-321-97710-6
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Dedicated to R. Kent Nagle
He has left his imprint not only on these pages but upon all who knew him. He was
that rare mathematician who could effectively communicate at all levels, imparting
his love for the subject with the same ease to undergraduates, graduates, precollege
students, public school teachers, and his colleagues at the University of South Florida.
Kent was at peace in life—a peace that emanated from the depth of his under-
standing of the human condition and the strength of his beliefs in the institutions of
family, religion, and education. He was a research mathematician, an accomplished
author, a Sunday school teacher, and a devoted husband and father.
Kent was also my dear friend and my jogging partner who has left me behind still
struggling to keep pace with his high ideals.
E. B. Saff
A01_NAGL7106_07_SE_FM_i-xx.indd 3 31/10/16 1:21 PM
