Table Of Content

REVIEW OF DIFFERENTIATION Rules d d 1. Constant: c=0 2. ConstantMultiple: cf(x)=cf(cid:0)(x) dx dx d d 3. Sum: [f(x) ± g(x)]=f(cid:0)(x)± g(cid:0)(x) 4. Product: f(x)g(x)= f(x)g(cid:0)(x) + g(x)f(cid:0)(x) dx dx d f(x) g(x)f(cid:0)(x)(cid:0) f(x)g(cid:0)(x) d 5. Quotient: = 6. Chain: f(g(x))= f(cid:0)(g(x))g(cid:0)(x) dx g(x) [g(x)]2 dx 7. Power: d xn =nxn(cid:0)1 8. Power: d [g(x)]n =n[g(x)]n(cid:0)1g(cid:0)(x) dx dx Functions Trigonometric: d d d 9. sinx=cosx 10. cosx=(cid:0)sinx 11. tanx=sec2 x dx dx dx d d d 12. cotx=(cid:0)csc2 x 13. secx=secxtanx 14. cscx=(cid:0)cscxcotx dx dx dx Inverse trigonometric: 15. d sin(cid:0)1 x= 1 16. d cos(cid:0)1 x=(cid:0) 1 17. d tan(cid:0)1 x= 1 dx 1(cid:0)x2 dx 1(cid:0)x2 dx 1+x2 18. d cot(cid:0)1 x=(cid:0) 1 19. d sec(cid:0)1 x= 1 20. d csc(cid:0)1 x=(cid:0) 1 dx 1+x2 dx x x2(cid:0)1 dx x x2(cid:0)1 Hyperbolic: d d d 21. sinhx=coshx 22. coshx=sinhx 23. tanhx=sech2 x dx dx dx d d d 24. cothx=(cid:0)csch2 x 25. sechx=(cid:0)sechxtanhx 26. cschx=(cid:0)cschxcothx dx dx dx Inverse hyperbolic: 27. d sinh(cid:0)1 x= 1 28. d cosh(cid:0)1 x= 1 29. d tanh(cid:0)1 x= 1 dx x2+1 dx x2(cid:0)1 dx 1(cid:0)x2 30. d coth(cid:0)1 x= 1 31. d sech(cid:0)1 x=(cid:0) 1 32. d csch(cid:0)1 x=(cid:0) 1 dx 1(cid:0)x2 dx x 1(cid:0)x2 dx x x2+1 Exponential: d d 33. ex =ex 34. bx =bx(lnb) dx dx Logarithmic: d 1 d 1 35. ln x = 36. log x= dx x dx b x(lnb) BRIEF TABLE OF INTEGRALS (cid:5) un(cid:1)1 (cid:5)1 1. undu(cid:2) (cid:1)C, n(cid:3)(cid:4)1 2. du(cid:2)ln u (cid:1)C n(cid:1)1 u (cid:5) (cid:5) 1 3. eudu(cid:2)eu (cid:1)C 4. audu(cid:2) au (cid:1)C lna (cid:5) (cid:5) 5. sin udu(cid:2)(cid:4)cos u(cid:1)C 6. cosudu(cid:2)sin u(cid:1)C (cid:5) (cid:5) 7. sec2udu(cid:2)tan u(cid:1)C 8. csc2udu(cid:2)(cid:4)cot u(cid:1)C (cid:5) (cid:5) 9. secutan udu(cid:2)secu(cid:1)C 10. cscucot udu(cid:2)(cid:4)cscu(cid:1)C (cid:5) (cid:5) 11. tan udu(cid:2)(cid:4)ln cos u (cid:1)C 12. cot udu(cid:2)ln sin u (cid:1)C (cid:5) (cid:5) 13. secudu(cid:2)ln secu(cid:1)tan u (cid:1)C 14. cscudu(cid:2)ln cscu(cid:4)cot u (cid:1)C (cid:5) (cid:5) 15. u sin udu(cid:2)sin u(cid:4)ucosu(cid:1)C 16. u cosudu(cid:2)cosu(cid:1) usin u(cid:1)C (cid:5) (cid:5) 17. sin2udu(cid:2) 1u(cid:4)1sin 2u (cid:1) C 18. cos2udu(cid:2) 1u(cid:1)1sin 2u (cid:1) C 2 4 2 4 (cid:5) (cid:5) 19. tan2udu(cid:2)tan u(cid:4)u (cid:1) C 20. cot2udu(cid:2)(cid:4)cot u(cid:4)u (cid:1) C (cid:5) (cid:6) (cid:7) (cid:5) (cid:6) (cid:7) 21. sin3udu(cid:2)(cid:4)1 2(cid:1)sin2u cosu (cid:1) C 22. cos3udu(cid:2) 1 2(cid:1)cos2u sin u (cid:1) C 3 3 (cid:5) (cid:5) 23. tan3udu(cid:2) 1tan2u(cid:1) ln cos u (cid:1) C 24. cot3udu(cid:2)(cid:4)1cot2u(cid:4) ln sin u (cid:1) C 2 2 (cid:5) (cid:5) 25. sec3udu(cid:2) 1secutanu(cid:1)1ln secu(cid:1)tanu (cid:1)C 26. csc3udu(cid:2)(cid:4)1cscucotu(cid:1)1ln cscu(cid:4)cotu (cid:1)C 2 2 2 2 (cid:5) sin(a(cid:4)b)u sin(a(cid:1)b)u (cid:5) sin(a(cid:4)b)u sin(a(cid:1)b)u 27. sin aucosbudu(cid:2) (cid:4) (cid:1) C 28. cos aucosbudu(cid:2) (cid:1) (cid:1) C 2(a(cid:4)b) 2(a(cid:1)b) 2(a(cid:4)b) 2(a(cid:1)b) 29. (cid:5)eausin budu(cid:2) eau (cid:6)asin bu(cid:4)bcosbu(cid:7)(cid:1)C 30. (cid:5)eaucos budu(cid:2) eau (cid:6)acosbu(cid:1)bsin bu(cid:7)(cid:1)C a2(cid:1)b2 a2(cid:1)b2 (cid:5) (cid:5) 31. sinh udu(cid:2)cosh u(cid:1)C 32. cosh udu(cid:2)sinh u(cid:1)C (cid:5) (cid:5) 33. sech2udu(cid:2)tanh u(cid:1)C 34. csch2udu(cid:2)(cid:4)coth u(cid:1)C (cid:5) (cid:5) 35. tanh udu(cid:2)ln(cosh u)(cid:1)C 36. coth udu(cid:2)ln sinh u (cid:1)C (cid:5) (cid:5) 37. ln udu(cid:2)ulnu(cid:4)u(cid:1)C 38. uln udu(cid:2) 1u2lnu(cid:4)1u2(cid:1)C 2 4 (cid:5) 1 u (cid:5) 1 39. du(cid:2)sin(cid:4)1 (cid:1)C 40. du(cid:2)ln u(cid:1) a2 (cid:1) u2 (cid:1)C a2(cid:4)u2 a a2(cid:1) u2 (cid:5) u a2 u (cid:5) u a2 41. a2(cid:4)u2 du(cid:2) a2(cid:4)u2 (cid:1) sin(cid:4)1 (cid:1)C 42. a2(cid:1) u2 du(cid:2) a2(cid:1) u2 (cid:1) ln u(cid:1) a2(cid:1) u2 (cid:1)C 2 2 a 2 2 (cid:5) 1 1 u (cid:5) 1 1 a(cid:1)u 43. du(cid:2) tan(cid:4)1 (cid:1) C 44. du(cid:2) ln (cid:1) C a2(cid:1)u2 a a a2(cid:4)u2 2a a(cid:4)u Note: Some techniques of integration, such as integration by parts and partial fractions, are reviewed in the Student Resource and Solutions Manual that accompanies this text. TABLE OF LAPLACE TRANSFORMS f(t) (cid:1){f(t)}(cid:7)F(s) f(t) (cid:1){f(t)}(cid:7)F(s) 1 k 1. 1 20. eatsinhkt s (s(cid:1)a)2(cid:1)k2 1 s(cid:1)a 2. t 21. eatcoshkt s2 (s(cid:1)a)2(cid:1)k2 n! 2ks 3. tn , na positive integer 22. tsinkt sn(cid:2)1 (s2(cid:2)k2)2 (cid:6) s2(cid:1)k2 4. t(cid:1)1/2 23. tcoskt Bs (s2(cid:2)k2)2 1(cid:6) 2ks2 5. t1/2 24. sinkt(cid:2)ktcoskt 2s3/2 (s2(cid:2)k2)2 (cid:3)((cid:4)(cid:2)1) 2k3 6. ta , a(cid:5)(cid:1)1 25. sinkt(cid:1)ktcoskt s(cid:4)(cid:2)1 (s2(cid:2)k2)2 k 2ks 7. sinkt 26. tsinhkt s2(cid:2)k2 (s2(cid:1)k2)2 s s2(cid:2)k2 8. coskt 27. tcoshkt s2(cid:2)k2 (s2(cid:1)k2)2 2k2 eat(cid:1)ebt 1 9. sin2kt 28. s(s2(cid:2)4k2) a(cid:1)b (s(cid:1)a)(s(cid:1)b) s2(cid:2)2k2 aeat(cid:1)bebt s 10. cos2kt 29. s(s2(cid:2)4k2) a(cid:1)b (s(cid:1)a)(s(cid:1)b) 1 k2 11. eat 30. 1(cid:1)coskt s(cid:1)a s(s2(cid:2)k2) k k3 12. sinhkt 31. kt(cid:1)sinkt s2(cid:1)k2 s2(s2(cid:2)k2) s a sin bt(cid:1)b sin at 1 13. coshkt 32. s2(cid:1)k2 ab(a2(cid:1)b2) (s2(cid:2)a2)(s2(cid:2)b2) 2k2 cosbt(cid:1)cosat s 14. sinh2kt 33. s(s2(cid:1)4k2) a2(cid:1)b2 (s2(cid:2)a2)(s2(cid:2)b2) s2(cid:1)2k2 2k2s 15. cosh2kt 34. sinktsinhkt s(s2(cid:1)4k2) s4(cid:2)4k4 1 k(s2(cid:2)2k2) 16. teat 35. sinktcoshkt (s(cid:1)a)2 s4(cid:2)4k4 n! k(s2(cid:1)2k2) 17. tneat , na positive integer 36. cosktsinhkt (s(cid:1)a)n(cid:2)1 s4(cid:2)4k4 k s3 18. eatsinkt 37. cosktcoshkt (s(cid:1)a)2(cid:2)k2 s4(cid:2)4k4 s(cid:1)a 1 19. eatcoskt 38. J (kt) (s(cid:1)a)2(cid:2)k2 0 1s2(cid:2)k2 This page intentionally left blank SEVENTH EDITION DIFFERENTIAL EQUATIONS with Boundary-Value Problems This page intentionally left blank SEVENTH EDITION DIFFERENTIAL EQUATIONS with Boundary-Value Problems DENNIS G. ZILL Loyola Marymount University MICHAEL R. 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Purchase any of our products at your local college store or at our preferred online store www.ichapters.com. Printed in Canada 1 2 3 4 5 6 7 12 11 10 09 08 CONTENTS Preface xi 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 1 1.1 Definitions and Terminology 2 1.2 Initial-Value Problems 13 1.3 Differential Equations as Mathematical Models 19 CHAPTER 1 IN REVIEW 32 2 FIRST-ORDER DIFFERENTIAL EQUATIONS 34 2.1 Solution Curves Without a Solution 35 2.1.1 Direction Fields 35 2.1.2 Autonomous First-Order DEs 37 2.2 Separable Variables 44 2.3 Linear Equations 53 2.4 Exact Equations 62 2.5 Solutions by Substitutions 70 2.6 A Numerical Method 75 CHAPTER 2 IN REVIEW 80 3 MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS 82 3.1 Linear Models 83 3.2 Nonlinear Models 94 3.3 Modeling with Systems ofFirst-Order DEs 105 CHAPTER 3 IN REVIEW 113 v