《微分方程》课程教学资源（书籍资料）DENNIS G. ZILL&MICHAEL R. CULLEN《Differential Equations with Boundary-Value Problems》（SEVENTH EDITION）

Differential Eguations With Boundary-Value Problems SEVENTH EDITION Dennis G.Zill Michael R.Cullen

REVIEW OF DIFFERENTIATION Rules 1.Constant: dc=0 2.Constant Multiple: f()=cf(x) d 3.Sum: f(田±gxl=f'e±gx 4.Product: d dx fxg)=f(g)+gefe 5.Quotient: dfx-gx)f"x)-fx)g'(） dx g(x) 6.Chain: [g(x2 足fgx》=fgxg'(x) 7.Power: d x”=nxn-1 8.Power: dx ng(g( dx Functions Trigonometric: 9. d d sinx=cosx 10. 11. dx dx cosx=-sinx d tanx=seex d 12.4cotx=-csc 13. d 14. dx -secx=sec xtanx cscx=-cscx cot Inverse trigonometric: 15d 1 16. d cos-1x=-- 1 17. dx 1-x2 dx V1-x2 dx 1+x2 18 ot-x=-1 -sec-1x=- 1 19. 20. d csc-1 x=-- 1 dx 1+x2 dx Ve2-1 dx Vx2-1 Hyperbolic: 21 sinhx=coshx 22. d -cosh x=sinhx 23. dx dx d tanhx=sech2x dx 24 -coth x=-csch2x 25. d sechx=-sech xtanhx 26. d -cschx =-csch x cothx dx dx dx Inverse hyperbolic: 27 -sinh-1x=- 1 28. d cosh-1x=- 1 29.d tanhx=1 dx x2+1 dx x2-1 1-x2 30. -coth=7 31. d 1-x2 sech-Ix=- d 32. csch-1x=- xv1-x2 dx Ve2+1 Exponential: mhvve 34.4b=banb) dx Logarithmic: 5县g=生 36. 1 dx d logx= d x(nb)

Rules 1. Constant: d dx c = 0 2. Constant Multiple: d dx cf (x) = c f ￾(x) . Sum: d dx [f (x) ± g(x)] = ￾ f (x) ± g￾(x) 4. Product: d dx f (x)g(x) = f (x)g￾(x) + g(x) f ￾(x) 5. Quotient: d dx f (x) g(x) = g(x)f (x) ￾ f (x)g (x) [ g(x)]2 6. Chain: d dx f (g(x)) = f ￾(g(x))g￾(x) 7. Power: d dx xn = nxn￾1 8. Power: d dx [ g(x)]n = n[ g(x)]n 1 g￾(x) Functions Trigonometric: 9. d dx sin x = cos x 10. d dx cos x = ￾ sin x 11. d dx tan x = sec2 x 12. d dx cot x = ￾ csc2 x 13. d dx sec x = sec x tan x 14. d dx csc x = ￾ csc x cot x Inverse trigonometric: 15. d dx sin￾1 x = 1 1 ￾ x2 16. d dx cos￾1 x = ￾ 1 1 ￾ x2 17. d dx tan￾1 x = 1 1 + x2 18. d dx cot￾1 x = ￾ 1 1 + x2 19. d dx sec￾1 x = 1 x x2 ￾1 20. d dx csc￾1 x = ￾ 1 x x2 ￾1 Hyperbolic: 21. d dx sinh x = cosh x 22. d dx cosh x = sinh x 23. d dx tanh x = sech2 x 24. d dx coth x = ￾ csch2 x 25. d dx sech x = ￾ sech x tanh x 26. d dx csch x = ￾ csch x coth x Inverse hyperbolic: 27. d dx sinh￾1 x = 1 x2 +1 28. d dx cosh￾1 x = 1 x2 ￾1 29. d dx tanh￾1 x = 1 1 ￾ x2 30. d dx coth￾1 x = 1 1 ￾ x2 31. d dx sech￾1 x = ￾ 1 x 1 ￾ x2 32. d dx csch￾1 x = ￾ 1 x x2 +1 Exponential: 33. d dx ex = ex 34. d dx bx = bx (lnb) Logarithmic: 35. d dx ln x = 1 x 36. d dx logb x = 1 x(lnb) ￾ ￾ ￾ 3 REVIEW OF DIFFERENTIATION

BRIEF TABLE OF INTEGRALS +C,n≠-1 n+1 2j片=p+c 3∫eda=c+c ∫as 4 a"+C Ina 5.sin udu =-cos u+C 6.cos udu=sin u+C 7. sec2 u du tan u+C 8 csci udu=-cot u+c 9.sec utan udu=sec u+C 10. csc u cot u du =-csc u+C 11.tan udu =-In cos u+C 12. cot udu =In sin u+C 13.sec u du Insec u+tan u+C 14.cse udu Inlesc u-cot u+C 15.u sin udu =sin u-ucosu+C 16.u cos u du cos u+usin u+C 17.sin'udu=tu-sin 2u+C 1 .cos'udu=u+sin u+c 19.tan'udu=tan u-u+C 20. cot2 udu =-cot u-u C 21.sin'udu=(2+sin'u)cosu+C 22. cos'udu=(2+cos'u)sin u+C 23.tan'udu =tan'u+In cos u+C 24 cot'udu =-+cot2u-Insin u+C 25. sec u du =isecu tanu+Insecu+tanu+C 26. cse u du =-+cscu cotu+Incscu-cotu+C 27. sin aucos bu du=sin(a-b)usin(abuC 28 cos au cos bu du-sin(absin(bc 2(a-b) 2(a+b) 2(a-b) 2(a+b) 29. [e"sin budu= e (asin bu-bcos bu)+C 30 e cos bu du = (acos bu+bsin bu)+C 31. sinh u du cosh u+C 32. cosh u du sinh u+C 33. sech2 u du tanh u+C 34. csch2 u du =-coth u+C 35. tanh u du In(cosh u)+C 36. coth u du Insinh u+C 37. In udu =ulnu-u+C 38. uln udu =iu2 Inu-iu2+C 39. a2- -du=sin+C 40. a ∫-=-r+ sin+C 42. ∫F+-F+++F++c 43. du=-tan+C 44. a2-= 1a+4+C Ja2+u a 2aa-u Note:Some techniques of integration,such as integration by parts and partial fractions,are reviewed in the Student Resource and Solutions Manal that accompanies this text

BRIEF TABLE OF INTEGRALS 1. 1 , 1 1 n n u u du C n n

 
2. 1 du u C ln u
 3. u u e du e C
 4. 1 ln u u a du a C a
 5. sin cos u du u C 
 6. cos sin u du u C
 7. 2 sec tan u du u C
 8. 2 csc cot u du u C 
 9. sec tan sec u u du u C
 10. csc cot csc u u du u C 
 11. tan ln cos u du u C 
 12. cot ln sin u du u C
 13. sec ln sec tan u du u u C

 14. csc ln csc cot u du u u C 
 15. u u du u u u C sin sin cos 
 16. u u du u u u C cos cos sin

 17. 2 1 1 2 4 sin sin 2 u du u u C 
 18. 2 1 1 2 4 cos sin 2 u du u u C

 19. 2 tan tan u du u u C 
 20. 2 cot cot u du u u C  
 21.   3 2 1 3 sin 2 sin cos u du u u C 

 22.   3 2 1 3 cos 2 cos sin u du u u C

 23. 3 2 1 2 tan tan ln cos u du u u C

 24. 3 2 1 2 cot cot ln sin u du u u C  
 25. 3 1 1 2 2 sec sec tan ln sec tan u du u u u u C


 26. 3 1 1 2 2 csc csc cot ln csc cot u du u u u u C 

 27. sin( ) sin( ) sin cos 2( ) 2( ) a bu a bu au bu du C ab ab 

 
28. sin( ) sin( ) cos cos 2( ) 2( ) a bu a bu au bu du C ab ab 


 
29.   2 2 sin sin cos au au e e bu du a bu b bu C a b 

30.   2 2 cos cos sin au au e e bu du a bu b bu C a b


31. sinh cosh u du u C
 32. cosh sinh u du u C
 33. 2 sech tanh u du u C
 34. 2 csch coth u du u C 
 35. tanh ln(cosh ) u du u C
 36. coth ln sinh u du u C
 37. ln ln u du u u u C 
 38. 1 1 2 2 2 4 u u du u u u C ln ln 
 39. 1 2 2 1 sin u du C a u a 
  40. 2 2 2 2 1 du u a u C ln a u



 41. 2 22 22 1 sin 2 2 u au a u du a u C a   

 42. 2 22 22 22 ln 2 2 u a a u du a u u a u C





 43. 1 2 2 1 1 tan u du C a u a a 

44. 2 2 1 1 ln 2 a u du C a u a au

   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(r) {f0=F) f() f())=F(s) 1.1 20.eat sinh k (s a)-K 21 21.ea cosh kt s-a 8-a2-2 3. n! napositive integer 22.tsin kt 2ks 2+2乎 4.1~2 限 23.tcoskt 2-k2 2+k2乎 5.72 温 24.sin kt kt cos kt 2k2 2+ 6. 25.sin kt-kt cos kt 2 G2+2乎 7.sin kt 2ks 2+2 26.t sinh kt 2-2呼 8.cos kt 27.t cosh kt s2+k2 2+2 (G2-k乎 22 28-山 1 9.sin2kt 2+4码 a-b (s-0s-b) 10.cos2kt 2+22 sG2+4k码 29.ae-beh a-b (s-a)(s -b) 1 2 11.eat s-a 30.1-cos kt s(2+码 3 12.sinh kt 2-R 3L.k红-sinkt 22+巧 13.cosh kt 32.asin br-bsin at ab(a2-b2) 2+232+b 14.sinh2kt 22 s62-4码 33.cos bi-cos at a2-b2 2+a2s2+b的 15.cosh2kr 32-2k2 34.sin kt sinh kr 23 s(2-4k码 ++40 1 16.tea 35.sin kt cosh kt k(2+2k2) 6- +4 17."eat n! 36.cos kt sinh kt k2-2 (s -ayi, n a positive integer s4+4 k 18.ett sin kt 6-a2+F 37.cos kt cosh kr +4 19.eat cos kt s-a G-a2+k及 38.Jo) 1 VF+元

TABLE OF LAPLACE TRANSFORMS f(t) 1. 1 2. t 3. t n n a positive integer 4. t
1/2 5. t 1/2 6. t a 7. sin kt 8. cos kt 9. sin2 kt 10. cos2 kt 11. eat 12. sinh kt 13. cosh kt 14. sinh2kt 15. cosh2kt 16. teat 17. t n eat n a positive integer 18. eat sin kt 19. eat cos kt s
a (s
a) 2 k2 k (s
a) 2 k2 n! (s
a) n1 , 1 (s
a) 2 s2
2k2 s(s2
4k2 ) 2k2 s(s2
4k2 ) s s2
k2 k s2
k2 1 s
a s2 2k2 s(s2 4k2 ) 2k2 s(s2 4k2 ) s s2 k2 k s2 k2 ( 1) s1 , a 
1 1 2s3/2 B  s n! sn1 , 1 s2 1 s
{ f (t)}  F(s) f(t) 20. eat sinh kt 21. eat cosh kt 22. t sin kt 23. t cos kt 24. sin kt kt cos kt 25. sin kt
kt cos kt 26. t sinh kt 27. t cosh kt 28. 29. 30. 1
cos kt 31. kt
sin kt 32. 33. 34. sin kt sinh kt 35. sin kt cosh kt 36. cos kt sinh kt 37. cos kt cosh kt 38. J0(kt) 1 1s2 k2 s3 s4 4k4 k(s2
2k2) s4 4k4 k(s2 2k2) s4 4k4 2k2 s s4 4k4 s (s2 a2 )(s2 b2 ) cos bt
cos at a2
b2 1 (s2 a2 )(s2 b2 ) a sin bt
b sin at ab (a2
b2 ) k3 s2(s2 k2 ) k2 s(s2 k2 ) s (s
a)(s
b) aeat
bebt a
b 1 (s
a)(s
b) eat
ebt a
b s2 k2 (s2
k2 ) 2 2ks (s2
k2 ) 2 2k3 (s2 k2 ) 2 2ks2 (s2 k2 ) 2 s2
k2 (s2 k2 ) 2 2ks (s2 k2 ) 2 s
a (s
a) 2
k2 k (s
a) 2
k2
{ f (t)}  F(s)

SEVENTH EDITION DIFFERENTIAL EQUATIONS with Boundary-Value Problems

SEVENTH EDITION DIFFERENTIAL EQUATIONS with Boundary-Value Problems

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 of First-Order DEs 105 CHAPTER 3 IN REVIEW 113

3 v CONTENTS 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 1 Preface xi 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 MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS 82 3.1 Linear Models 83 3.2 Nonlinear Models 94 3.3 Modeling with Systems of First-Order DEs 105 CHAPTER 3 IN REVIEW 113

i ·CONTENTS 4 HIGHER-ORDER DIFFERENTIAL EQUATIONS 117 4.1 Preliminary Theory-Linear Equations 118 4.1.1 Initial-Value and Boundary-Value Problems 118 4.1.2 Homogeneous Equations 120 4.1.3 Nonhomogeneous Equations 125 4.2 Reduction of Order 130 4.3 Homogeneous Linear Equations with Constant Coefficients 133 4.4 Undetermined Coefficients-Superposition Approach 140 4.5 Undetermined Coefficients-Annihilator Approach 150 4.6 Variation of Parameters 157 4.7 Cauchy-Euler Equation 162 4.8 Solving Systems of Linear DEs by Elimination 169 4.9 Nonlinear Differential Equations 174 CHAPTER 4 IN REVIEW 178 5 MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS 181 5.1 Linear Models:Initial-Value Problems 182 5.1.1 Spring/Mass Systems:Free Undamped Motion 182 5.1.2 Spring/Mass Systems:Free Damped Motion 186 5.1.3 Spring/Mass Systems:Driven Motion 189 5.1.4 Series Circuit Analogue 192 5.2 Linear Models:Boundary-Value Problems 199 5.3 Nonlinear Models 207 CHAPTER 5 IN REVIEW 216 6 SERIES SOLUTIONS OF LINEAR EQUATIONS 219 6.1 Solutions About Ordinary Points 220 6.1.1 Review of Power Series 220 6.1.2 Power Series Solutions 223 6.2 Solutions About Singular Points 231 6.3 Special Functions 241 6.3.1 Bessel's Equation 241 6.3.2 Legendre's Equation 248 CHAPTER 6 IN REVIEW 253

5 4 vi ● CONTENTS HIGHER-ORDER DIFFERENTIAL EQUATIONS 117 4.1 Preliminary Theory—Linear Equations 118 4.1.1 Initial-Value and Boundary-Value Problems 118 4.1.2 Homogeneous Equations 120 4.1.3 Nonhomogeneous Equations 125 4.2 Reduction of Order 130 4.3 Homogeneous Linear Equations with Constant Coefficients 133 4.4 Undetermined Coefficients—Superposition Approach 140 4.5 Undetermined Coefficients—Annihilator Approach 150 4.6 Variation of Parameters 157 4.7 Cauchy-Euler Equation 162 4.8 Solving Systems of Linear DEs by Elimination 169 4.9 Nonlinear Differential Equations 174 CHAPTER 4 IN REVIEW 178 MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS 181 5.1 Linear Models: Initial-Value Problems 182 5.1.1 Spring/Mass Systems: Free Undamped Motion 182 5.1.2 Spring/Mass Systems: Free Damped Motion 186 5.1.3 Spring/Mass Systems: Driven Motion 189 5.1.4 Series Circuit Analogue 192 5.2 Linear Models: Boundary-Value Problems 199 5.3 Nonlinear Models 207 CHAPTER 5 IN REVIEW 216 SERIES SOLUTIONS OF LINEAR EQUATIONS 219 6.1 Solutions About Ordinary Points 220 6.1.1 Review of Power Series 220 6.1.2 Power Series Solutions 223 6.2 Solutions About Singular Points 231 6.3 Special Functions 241 6.3.1 Bessel’s Equation 241 6.3.2 Legendre’s Equation 248 CHAPTER 6 IN REVIEW 253 6

CONTENTS vii 7 THE LAPLACE TRANSFORM 255 7.1 Definition of the Laplace Transform 256 7.2 Inverse Transforms and Transforms of Derivatives 262 7.2.1 Inverse Transforms 262 7.2.2 Transforms of Derivatives 265 7.3 Operational Properties I 270 7.3.1 Translation on the s-Axis 271 7.3.2 Translation on the t-Axis 274 7.4 Operational Properties II 282 7.4.1 Derivatives of a Transform 282 7.4.2 Transforms of Integrals 283 7.4.3 Transform of a Periodic Function 287 7.5 The Dirac Delta Function 292 7.6 Systems of Linear Differential Equations 295 CHAPTER 7 IN REVIEW 300 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS 303 8.1 Preliminary Theory-Linear Systems 304 8.2 Homogeneous Linear Systems 311 8.2.1 Distinct Real Eigenvalues 312 8.2.2 Repeated Eigenvalues 315 8.2.3 Complex Eigenvalues 320 8.3 Nonhomogeneous Linear Systems 326 8.3.1 Undetermined Coefficients 326 8.3.2 Variation of Parameters 329 8.4 Matrix Exponential 334 CHAPTER 8 IN REVIEW 337 9 NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS 339 9.1 Euler Methods and Error Analysis 340 9.2 Runge-Kutta Methods 345 9.3 Multistep Methods 350 9.4 Higher-Order Equations and Systems 353 9.5 Second-Order Boundary-Value Problems 358 CHAPTER 9 IN REVIEW 362

CONTENTS ● vii 7 THE LAPLACE TRANSFORM 255 7.1 Definition of the Laplace Transform 256 7.2 Inverse Transforms and Transforms of Derivatives 262 7.2.1 Inverse Transforms 262 7.2.2 Transforms of Derivatives 265 7.3 Operational Properties I 270 7.3.1 Translation on the s-Axis 271 7.3.2 Translation on the t-Axis 274 7.4 Operational Properties II 282 7.4.1 Derivatives of a Transform 282 7.4.2 Transforms of Integrals 283 7.4.3 Transform of a Periodic Function 287 7.5 The Dirac Delta Function 292 7.6 Systems of Linear Differential Equations 295 CHAPTER 7 IN REVIEW 300 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS 303 8.1 Preliminary Theory—Linear Systems 304 8.2 Homogeneous Linear Systems 311 8.2.1 Distinct Real Eigenvalues 312 8.2.2 Repeated Eigenvalues 315 8.2.3 Complex Eigenvalues 320 8.3 Nonhomogeneous Linear Systems 326 8.3.1 Undetermined Coefficients 326 8.3.2 Variation of Parameters 329 8.4 Matrix Exponential 334 CHAPTER 8 IN REVIEW 337 9 NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS 339 9.1 Euler Methods and Error Analysis 340 9.2 Runge-Kutta Methods 345 9.3 Multistep Methods 350 9.4 Higher-Order Equations and Systems 353 9.5 Second-Order Boundary-Value Problems 358 CHAPTER 9 IN REVIEW 362

viii CONTENTS 10 PLANE AUTONOMOUS SYSTEMS 363 10.1 Autonomous Systems 364 10.2 Stability of Linear Systems 370 10.3 Linearization and Local Stability 378 10.4 Autonomous Systems as Mathematical Models 388 CHAPTER 10 IN REVIEW 395 11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES 397 11.1 Orthogonal Functions 398 11.2 Fourier Series 403 11.3 Fourier Cosine and Sine Series 408 11.4 Sturm-Liouville Problem 416 11.5 Bessel and Legendre Series 423 11.5.1 Fourier-Bessel Series 424 11.5.2 Fourier-Legendre Series 427 CHAPTER 11 IN REVIEW 430 12 BOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES 432 12.1 Separable Partial Differential Equations 433 12.2 Classical PDEs and Boundary-Value Problems 437 12.3 Heat Equation 443 12.4 Wave Equation 445 12.5 Laplace's Equation 450 12.6 Nonhomogeneous Boundary-Value Problems 455 12.7 Orthogonal Series Expansions 461 12.8 Higher-Dimensional Problems 466 CHAPTER 12 IN REVIEW 469

viii ● CONTENTS 10 PLANE AUTONOMOUS SYSTEMS 363 10.1 Autonomous Systems 364 10.2 Stability of Linear Systems 370 10.3 Linearization and Local Stability 378 10.4 Autonomous Systems as Mathematical Models 388 CHAPTER 10 IN REVIEW 395 11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES 397 11.1 Orthogonal Functions 398 11.2 Fourier Series 403 11.3 Fourier Cosine and Sine Series 408 11.4 Sturm-Liouville Problem 416 11.5 Bessel and Legendre Series 423 11.5.1 Fourier-Bessel Series 424 11.5.2 Fourier-Legendre Series 427 CHAPTER 11 IN REVIEW 430 12 BOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES 432 12.1 Separable Partial Differential Equations 433 12.2 Classical PDEs and Boundary-Value Problems 437 12.3 Heat Equation 443 12.4 Wave Equation 445 12.5 Laplace’s Equation 450 12.6 Nonhomogeneous Boundary-Value Problems 455 12.7 Orthogonal Series Expansions 461 12.8 Higher-Dimensional Problems 466 CHAPTER 12 IN REVIEW 469

CONTENTS 。i 13 BOUNDARY-VALUE PROBLEMS IN OTHER COORDINATE SYSTEMS 471 13.1 Polar Coordinates 472 13.2 Polar and Cylindrical Coordinates 477 13.3 Spherical Coordinates 483 CHAPTER 13 IN REVIEW 486 14 INTEGRAL TRANSFORMS 488 14.1 Error Function 489 14.2 Laplace Transform 490 14.3 Fourier Integral 498 14.4 Fourier Transforms 504 CHAPTER 14 IN REVIEW 510 15 NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS 511 15.1 Laplace's Equation 512 15.2 Heat Equation 517 15.3 Wave Equation 522 CHAPTER 15 IN REVIEW 526 APPENDICES Gamma Function APP-1 I Matrices APP-3 Laplace Transforms APP-21 Answers for Selected Odd-Numbered Problems ANS-1 Index I-1

CONTENTS ● ix 13 BOUNDARY-VALUE PROBLEMS IN OTHER COORDINATE SYSTEMS 471 13.1 Polar Coordinates 472 13.2 Polar and Cylindrical Coordinates 477 13.3 Spherical Coordinates 483 CHAPTER 13 IN REVIEW 486 14 INTEGRAL TRANSFORMS 488 14.1 Error Function 489 14.2 Laplace Transform 490 14.3 Fourier Integral 498 14.4 Fourier Transforms 504 CHAPTER 14 IN REVIEW 510 15 NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS 511 15.1 Laplace’s Equation 512 15.2 Heat Equation 517 15.3 Wave Equation 522 CHAPTER 15 IN REVIEW 526 APPENDICES I Gamma Function APP-1 II Matrices APP-3 III Laplace Transforms APP-21 Answers for Selected Odd-Numbered Problems ANS-1 Index I-1