Instructor’s Solutions Manual
for
Differential Equations and
Linear Algebra, 4th edition,
ST
By C. Edwards, David Penney
U
D
YL
(All Chapters 1-11, 100%
AB
Original Verified, A+ Grade)
, Table of Content
First-Order Differential Equations
1.1 Differential Equations and Mathematical Models
1.2 Integrals as General and Particular Solutions
1.3 Slope Fields and Solution Curves
1.4 Separable Equations and Applications
1.5 Linear First-Order Equations
1.6 Substitution Methods and Exact Equations
Mathematical Models and Numerical Methods
2.1 Population Models
2.2 Equilibrium Solutions and Stability
2.3 Acceleration - Velocity Models
2.4 Numerical Approximation: Euler's Method
2.5 A Closer Look at the Euler Method
ST
2.6 The Runge - Kutta Method
Linear Systems and Matrices
3.1 Introduction to Linear Systems
3.2 Matrices and Gaussian Elimination
3.3 Reduced Row-Echelon Matrices
U
3.4 Matrix Operations
3.5 Inverses of Matrices
D
3.6 Determinants
3.7 Linear Equations and Curve Fitting
Vector Spaces
YL
4.1 The Vector Space R3
4.2 The Vector Space Rn and Subspaces
4.3 Linear Combinations and Independence of Vectors
4.4 Bases and Dimension for Vector Spaces
AB
4.5 Row and Column Spaces
4.6 Orthogonal Vectors in Rn
4.7 General Vector Spaces
Higher-Order Linear Differential Equations
5.1 Introduction: Second-Order Linear Equations
5.2 General Solutions of Linear Equations
5.3 Homogeneous Equations with Constant Coefficients
5.4 Mechanical Vibrations
5.5 Nonhomogeneous Equations and Undetermined Coefficients
5.6 Forced Oscillations and Resonance
Eigenvalues and Eigenvectors
6.1 Introduction to Eigenvalues
6.2 Diagonalization of Matrices
6.3 Applications Involving Powers of Matrices
Linear Systems of Differential Equations
,7.1 First-Order Systems and Applications
7.2 Matrices and Linear Systems
7.3 The Eigenvalue Method for Linear Systems
7.4 A Gallery of Solution Curves of Linear Systems
7.5 Second-Order Systems and Mechanical Applications
7.6 Multiple Eigenvalue Solutions
7.7 Numerical Methods for Systems
Matrix Exponential Methods
8.1 Matrix Exponentials and Linear Systems
8.2 Nonhomogeneous Linear Systems
8.3 Spectral Decomposition Methods
Nonlinear Systems and Phenomena
9.1 Stability and the Phase Plane
9.2 Linear and Almost Linear Systems
ST
9.3 Ecological Models: Predators and Competitors
9.4 Nonlinear Mechanical Systems
Laplace Transform Methods
10.1 Laplace Transforms and Inverse Transforms
10.2 Transformation of Initial Value Problems
U
10.3 Translation and Partial Fractions
10.4 Derivatives, Integrals, and Products of Transforms
D
10.5 Periodic and Piecewise Continuous Input Functions
Power Series Methods
11.1 Introduction and Review of Power Series
YL
11.2 Power Series Solutions
11.3 Frobenius Series Solutions
11.4 Bessel Functions
AB
, CHAPTER 1
FIRST-ORDER DIFFERENTIAL EQUATIONS
SECTION 1.1
DIFFERENTIAL EQUATIONS AND MATHEMATICAL MODELS
The main purpose of Section 1.1 is simply to introduce the basic notation and terminology of dif-
ferential equations, and to show the student what is meant by a solution of a differential equation.
Also, the use of differential equations in the mathematical modeling of real-world phenomena is
outlined.
ST
Problems 1-12 are routine verifications by direct substitution of the suggested solutions into the
given differential equations. We include here just some typical examples of such verifications.
3. If y1 cos 2 x and y2 sin 2 x , then y1 2sin 2 x y2 2 cos 2 x , so
U
y1 4 cos 2 x 4 y1 and y2 4sin 2 x 4 y2 . Thus y1 4 y1 0 and y2 4 y2 0 .
4. If y1 e 3 x and y 2 e 3 x , then y1 3 e3 x and y2 3 e 3 x , so y1 9e 3 x 9 y1 and
D
y2 9e 3 x 9 y2 .
If y e x e x , then y e x e x , so y y e x e x e x e x 2 e x . Thus
YL
5.
y y 2 e x .
6. If y1 e 2 x and y2 x e 2 x , then y1 2 e 2 x , y1 4 e 2 x , y 2 e 2 x 2 x e 2 x , and
AB
y 2 4 e 2 x 4 x e 2 x . Hence
y1 4 y1 4 y1 4 e 2 x 4 2 e 2 x 4 e 2 x 0
and
y2 4 y2 4 y2 4e 2 x
4 x e 2 x 4 e 2 x 2 x e 2 x 4 x e 2 x 0.
8. If y1 cos x cos 2 x and y2 sin x cos 2 x , then y1 sin x 2sin 2 x,
y1 cos x 4 cos 2 x, y2 cos x 2sin 2 x , and y2 sin x 4 cos 2 x. Hence
y1 y1 cos x 4 cos 2 x cos x cos 2 x 3cos 2 x
and
y2 y2 sin x 4cos 2 x sin x cos 2 x 3cos 2 x.
1
Copyright © 2018 Pearson Education, Inc.
for
Differential Equations and
Linear Algebra, 4th edition,
ST
By C. Edwards, David Penney
U
D
YL
(All Chapters 1-11, 100%
AB
Original Verified, A+ Grade)
, Table of Content
First-Order Differential Equations
1.1 Differential Equations and Mathematical Models
1.2 Integrals as General and Particular Solutions
1.3 Slope Fields and Solution Curves
1.4 Separable Equations and Applications
1.5 Linear First-Order Equations
1.6 Substitution Methods and Exact Equations
Mathematical Models and Numerical Methods
2.1 Population Models
2.2 Equilibrium Solutions and Stability
2.3 Acceleration - Velocity Models
2.4 Numerical Approximation: Euler's Method
2.5 A Closer Look at the Euler Method
ST
2.6 The Runge - Kutta Method
Linear Systems and Matrices
3.1 Introduction to Linear Systems
3.2 Matrices and Gaussian Elimination
3.3 Reduced Row-Echelon Matrices
U
3.4 Matrix Operations
3.5 Inverses of Matrices
D
3.6 Determinants
3.7 Linear Equations and Curve Fitting
Vector Spaces
YL
4.1 The Vector Space R3
4.2 The Vector Space Rn and Subspaces
4.3 Linear Combinations and Independence of Vectors
4.4 Bases and Dimension for Vector Spaces
AB
4.5 Row and Column Spaces
4.6 Orthogonal Vectors in Rn
4.7 General Vector Spaces
Higher-Order Linear Differential Equations
5.1 Introduction: Second-Order Linear Equations
5.2 General Solutions of Linear Equations
5.3 Homogeneous Equations with Constant Coefficients
5.4 Mechanical Vibrations
5.5 Nonhomogeneous Equations and Undetermined Coefficients
5.6 Forced Oscillations and Resonance
Eigenvalues and Eigenvectors
6.1 Introduction to Eigenvalues
6.2 Diagonalization of Matrices
6.3 Applications Involving Powers of Matrices
Linear Systems of Differential Equations
,7.1 First-Order Systems and Applications
7.2 Matrices and Linear Systems
7.3 The Eigenvalue Method for Linear Systems
7.4 A Gallery of Solution Curves of Linear Systems
7.5 Second-Order Systems and Mechanical Applications
7.6 Multiple Eigenvalue Solutions
7.7 Numerical Methods for Systems
Matrix Exponential Methods
8.1 Matrix Exponentials and Linear Systems
8.2 Nonhomogeneous Linear Systems
8.3 Spectral Decomposition Methods
Nonlinear Systems and Phenomena
9.1 Stability and the Phase Plane
9.2 Linear and Almost Linear Systems
ST
9.3 Ecological Models: Predators and Competitors
9.4 Nonlinear Mechanical Systems
Laplace Transform Methods
10.1 Laplace Transforms and Inverse Transforms
10.2 Transformation of Initial Value Problems
U
10.3 Translation and Partial Fractions
10.4 Derivatives, Integrals, and Products of Transforms
D
10.5 Periodic and Piecewise Continuous Input Functions
Power Series Methods
11.1 Introduction and Review of Power Series
YL
11.2 Power Series Solutions
11.3 Frobenius Series Solutions
11.4 Bessel Functions
AB
, CHAPTER 1
FIRST-ORDER DIFFERENTIAL EQUATIONS
SECTION 1.1
DIFFERENTIAL EQUATIONS AND MATHEMATICAL MODELS
The main purpose of Section 1.1 is simply to introduce the basic notation and terminology of dif-
ferential equations, and to show the student what is meant by a solution of a differential equation.
Also, the use of differential equations in the mathematical modeling of real-world phenomena is
outlined.
ST
Problems 1-12 are routine verifications by direct substitution of the suggested solutions into the
given differential equations. We include here just some typical examples of such verifications.
3. If y1 cos 2 x and y2 sin 2 x , then y1 2sin 2 x y2 2 cos 2 x , so
U
y1 4 cos 2 x 4 y1 and y2 4sin 2 x 4 y2 . Thus y1 4 y1 0 and y2 4 y2 0 .
4. If y1 e 3 x and y 2 e 3 x , then y1 3 e3 x and y2 3 e 3 x , so y1 9e 3 x 9 y1 and
D
y2 9e 3 x 9 y2 .
If y e x e x , then y e x e x , so y y e x e x e x e x 2 e x . Thus
YL
5.
y y 2 e x .
6. If y1 e 2 x and y2 x e 2 x , then y1 2 e 2 x , y1 4 e 2 x , y 2 e 2 x 2 x e 2 x , and
AB
y 2 4 e 2 x 4 x e 2 x . Hence
y1 4 y1 4 y1 4 e 2 x 4 2 e 2 x 4 e 2 x 0
and
y2 4 y2 4 y2 4e 2 x
4 x e 2 x 4 e 2 x 2 x e 2 x 4 x e 2 x 0.
8. If y1 cos x cos 2 x and y2 sin x cos 2 x , then y1 sin x 2sin 2 x,
y1 cos x 4 cos 2 x, y2 cos x 2sin 2 x , and y2 sin x 4 cos 2 x. Hence
y1 y1 cos x 4 cos 2 x cos x cos 2 x 3cos 2 x
and
y2 y2 sin x 4cos 2 x sin x cos 2 x 3cos 2 x.
1
Copyright © 2018 Pearson Education, Inc.
