Student Solutions Manual for Elementary Differential Equations and
Elementary Differential Equations with Boundary Value Problems
.
TO BEVERLY
Contents
Chapter 1 Introduction 1
1.2 First Order Equations 1
Chapter 2 First Order Equations 5
2.1 Linear First Order Equations 5
2.2 Separable Equations 8
2.3 Existence and Uniqueness of Solutions of Nonlinear Equations 11
2.4 Transformation of Nonlinear Equations into Separable Equations 13
2.5 Exact Equations 17
2.6 Integrating Factors 21
Chapter 3 Numerical Methods 25
3.1 Euler’s Method 25
3.2 The Improved Euler Method and Related Methods 29
,ii Contents
3.3 The Runge-Kutta Method 34
Chapter 4 Applications of First Order Equations 39
4.1 Growth and Decay 39
4.2 Cooling and Mixing 40
4.3 Elementary Mechanics 43
4.4 Autonomous Second Order Equations 45
4.5 Applications to Curves 46
Chapter 5 Linear Second Order Equations 51
5.1 Homogeneous Linear Equations 51
5.2 Constant Coefficient Homogeneous Equations 55
5.3 Nonhomgeneous Linear Equations 58
5.4 The Method of Undetermined Coefficients I 60
5.5 The Method of Undetermined Coefficients II 64
5.6 Reduction of Order 75
5.7 Variation of Parameters 79
Chapter 6 Applcations of Linear Second Order Equations 85
6.1 Spring Problems I 85
6.2 Spring Problems II 87
6.3 The RLC Circuit 89
6.4 Motion Under a Central Force 90
Chapter 7 Series Solutions of Linear Second Order Equations 108
7.1 Review of Power Series 91
7.2 Series Solutions Near an Ordinary Point I 93
7.3 Series Solutions Near an Ordinary Point II 96
7.4 Regular Singular Points; Euler Equations 102
7.5 The Method of Frobenius I 103
7.6 The Method of Frobenius II 108
7.7 The Method of Frobenius III 118
Chapter 8 Laplace Transforms 125
8.1 Introduction to the Laplace Transform 125
8.2 The Inverse Laplace Transform 127
8.3 Solution of Initial Value Problems 134
8.4 The Unit Step Function 140
8.5 Constant Coefficient Equations with Piecewise Continuous Forcing
Functions 143
8.6 Convolution 152
,Contents iii
8.7 Constant Cofficient Equations with Impulses 55
Chapter 9 Linear Higher Order Equations 159
9.1 Introduction to Linear Higher Order Equations 159
9.2 Higher Order Constant Coefficient Homogeneous Equations 171
9.3 Undetermined Coefficients for Higher Order Equations 175
9.4 Variation of Parameters for Higher Order Equations 181
Chapter 10 Linear Systems of Differential Equations 221
10.1 Introduction to Systems of Differential Equations 191
10.2 Linear Systems of Differential Equations 192
10.3 Basic Theory of Homogeneous Linear Systems 193
10.4 Constant Coefficient Homogeneous Systems I 194
10.5 Constant Coefficient Homogeneous Systems II 201
10.6 Constant Coefficient Homogeneous Systems II 245
10.7 Variation of Parameters for Nonhomogeneous Linear Systems 218
Chapter 221
11.1 Eigenvalue Problems for yrr hy 0 221
11.2 Fourier Expansions I 223
11.3 Fourier Expansions II 229
Chapter 12 Fourier Solutions of Partial Differential Equations 239
12.1 The Heat Equation 239
12.2 The Wave Equation 247
12.3 Laplace’s Equation in Rectangular Coordinates 260
12.4 Laplace’s Equation in Polar Coordinates 270
Chapter 13 Boundary Value Problems for Second Order Ordinary Differential Equations 273
13.1 Two-Point Boundary Value Problems 273
13.2 Sturm-Liouville Problems 279
, CHAPTER 1
Introduction
1.2 BASIC CONCEPTS
1.2.2. (a) If y 2= ce2x, then yr = 2ce2x = 2y.
x c 2x c 2x2 c x2 c 2
r r
(b) If y = 3 + , then y = — 2
, so xy + y = — + + =x .
x 3 x 3 x 3 x
(c) If
1
y = + ce—x , then yr = —2xce—x
2 2
2
and
1
yr + 2xy = —2xce—x + 2x + ce—x = —2xce—x + x + 2cxe—x = x.
2 2 2 2
2
(d) If
ce —
2
1
y=
1 — ce—x2/2
then
2 2 2 2
yr
= (1 — ce—x /2
)(—cxe—x /2) — (1 + ce—x )cxe—x
/2 /2
(1 — cxe—x /2)2
2
= —2cxe—x /2
(1 — ce—x /2)2
2
and
2
!2
1 + ce—x /2
y2 — 1 = 1
1 — ce—x /2
2 2
(1 + ce—x /2)2 — (1 — ce—x /2)2
= 2
(1 — ce—x /2)2
4ce—x /2
= ,
(1 — ce—x /2)2
2
Elementary Differential Equations with Boundary Value Problems
.
TO BEVERLY
Contents
Chapter 1 Introduction 1
1.2 First Order Equations 1
Chapter 2 First Order Equations 5
2.1 Linear First Order Equations 5
2.2 Separable Equations 8
2.3 Existence and Uniqueness of Solutions of Nonlinear Equations 11
2.4 Transformation of Nonlinear Equations into Separable Equations 13
2.5 Exact Equations 17
2.6 Integrating Factors 21
Chapter 3 Numerical Methods 25
3.1 Euler’s Method 25
3.2 The Improved Euler Method and Related Methods 29
,ii Contents
3.3 The Runge-Kutta Method 34
Chapter 4 Applications of First Order Equations 39
4.1 Growth and Decay 39
4.2 Cooling and Mixing 40
4.3 Elementary Mechanics 43
4.4 Autonomous Second Order Equations 45
4.5 Applications to Curves 46
Chapter 5 Linear Second Order Equations 51
5.1 Homogeneous Linear Equations 51
5.2 Constant Coefficient Homogeneous Equations 55
5.3 Nonhomgeneous Linear Equations 58
5.4 The Method of Undetermined Coefficients I 60
5.5 The Method of Undetermined Coefficients II 64
5.6 Reduction of Order 75
5.7 Variation of Parameters 79
Chapter 6 Applcations of Linear Second Order Equations 85
6.1 Spring Problems I 85
6.2 Spring Problems II 87
6.3 The RLC Circuit 89
6.4 Motion Under a Central Force 90
Chapter 7 Series Solutions of Linear Second Order Equations 108
7.1 Review of Power Series 91
7.2 Series Solutions Near an Ordinary Point I 93
7.3 Series Solutions Near an Ordinary Point II 96
7.4 Regular Singular Points; Euler Equations 102
7.5 The Method of Frobenius I 103
7.6 The Method of Frobenius II 108
7.7 The Method of Frobenius III 118
Chapter 8 Laplace Transforms 125
8.1 Introduction to the Laplace Transform 125
8.2 The Inverse Laplace Transform 127
8.3 Solution of Initial Value Problems 134
8.4 The Unit Step Function 140
8.5 Constant Coefficient Equations with Piecewise Continuous Forcing
Functions 143
8.6 Convolution 152
,Contents iii
8.7 Constant Cofficient Equations with Impulses 55
Chapter 9 Linear Higher Order Equations 159
9.1 Introduction to Linear Higher Order Equations 159
9.2 Higher Order Constant Coefficient Homogeneous Equations 171
9.3 Undetermined Coefficients for Higher Order Equations 175
9.4 Variation of Parameters for Higher Order Equations 181
Chapter 10 Linear Systems of Differential Equations 221
10.1 Introduction to Systems of Differential Equations 191
10.2 Linear Systems of Differential Equations 192
10.3 Basic Theory of Homogeneous Linear Systems 193
10.4 Constant Coefficient Homogeneous Systems I 194
10.5 Constant Coefficient Homogeneous Systems II 201
10.6 Constant Coefficient Homogeneous Systems II 245
10.7 Variation of Parameters for Nonhomogeneous Linear Systems 218
Chapter 221
11.1 Eigenvalue Problems for yrr hy 0 221
11.2 Fourier Expansions I 223
11.3 Fourier Expansions II 229
Chapter 12 Fourier Solutions of Partial Differential Equations 239
12.1 The Heat Equation 239
12.2 The Wave Equation 247
12.3 Laplace’s Equation in Rectangular Coordinates 260
12.4 Laplace’s Equation in Polar Coordinates 270
Chapter 13 Boundary Value Problems for Second Order Ordinary Differential Equations 273
13.1 Two-Point Boundary Value Problems 273
13.2 Sturm-Liouville Problems 279
, CHAPTER 1
Introduction
1.2 BASIC CONCEPTS
1.2.2. (a) If y 2= ce2x, then yr = 2ce2x = 2y.
x c 2x c 2x2 c x2 c 2
r r
(b) If y = 3 + , then y = — 2
, so xy + y = — + + =x .
x 3 x 3 x 3 x
(c) If
1
y = + ce—x , then yr = —2xce—x
2 2
2
and
1
yr + 2xy = —2xce—x + 2x + ce—x = —2xce—x + x + 2cxe—x = x.
2 2 2 2
2
(d) If
ce —
2
1
y=
1 — ce—x2/2
then
2 2 2 2
yr
= (1 — ce—x /2
)(—cxe—x /2) — (1 + ce—x )cxe—x
/2 /2
(1 — cxe—x /2)2
2
= —2cxe—x /2
(1 — ce—x /2)2
2
and
2
!2
1 + ce—x /2
y2 — 1 = 1
1 — ce—x /2
2 2
(1 + ce—x /2)2 — (1 — ce—x /2)2
= 2
(1 — ce—x /2)2
4ce—x /2
= ,
(1 — ce—x /2)2
2
