Differential Equations Computing and Modeling 5th Edition Edwards Solutions Manual

Differential Equations Computing and Modeling
5th Edition Edwards Solutions Manual

, Table contents:
1. 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

2. 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

▪ 2.6 The Runge—Kutta Method

3. Linear Equations of Higher Order

▪ 3.1 Introduction: Second-Order Linear Equations

▪ 3.2 General Solutions of Linear Equations

▪ 3.3 Homogeneous Equations with Constant Coefficients

▪ 3.4 Mechanical Vibrations

▪ 3.5 Nonhomogeneous Equations and Undetermined Coefficients

▪ 3.6 Forced Oscillations and Resonance

▪ 3.7 Electrical Circuits

▪ 3.8 Endpoint Problems and Eigenvalues

4. Introduction to Systems of Differential Equations

, ▪ 4.1 First-Order Systems and Applications

▪ 4.2 The Method of Elimination

▪ 4.3 Numerical Methods for Systems

5. Linear Systems of Differential Equations

▪ 5.1 Matrices and Linear Systems

▪ 5.2 The Eigenvalue Method for Homogeneous Systems

▪ 5.3 A Gallery of Solution Curves of Linear Systems

▪ 5.4 Second-Order Systems and Mechanical Applications

▪ 5.5 Multiple Eigenvalue Solutions

▪ 5.6 Matrix Exponentials and Linear Systems

▪ 5.7 Nonhomogeneous Linear Systems

6. Nonlinear Systems and Phenomena

▪ 6.1 Stability and the Phase Plane

▪ 6.2 Linear and Almost Linear Systems

▪ 6.3 Ecological Models: Predators and Competitors

▪ 6.4 Nonlinear Mechanical Systems

▪ 6.5 Chaos in Dynamical Systems

7. Laplace Transform Methods

▪ 7.1 Laplace Transforms and Inverse Transforms

▪ 7.2 Transformation of Initial Value Problems

▪ 7.3 Translation and Partial Fractions

▪ 7.4 Derivatives, Integrals, and Products of Transforms

▪ 7.5 Periodic and Piecewise Continuous Input Functions

▪ 7.6 Impulses and Delta Functions

, CHAPTER 1 g.




FIRST-ORDER DIFFERENTIAL EQUATIONS g. g.




SECTION 1.1 g.




DIFFERENTIAL EQUATIONS AND MATHEMATICAL MODELS g. g. g. g.




The main purpose of Section 1.1 is simply to introduce the basic notation and
g. g. g. g. g. g. g. g. g. g. g. g. g.



g. terminology of dif-ferential equations, and to show the student what is meant by
g. g. g. g. g. g. g. g. g. g. g. g.



g. a solution of a differential equation. Also, the use of differential equations in
g. g. g. g. g. g. g. g. g. g. g. g.



g. the mathematical modeling of real-world phenomena is outlined.
g. g. g. g. g. g. g.




Problems 1-12 are routine verifications by direct substitution of the suggested
g. g. g. g. g. g. g. g. g. g.



solutions into the given differential equations. We include here just some
g. g. g. g. g. g. g. g. g. g. g.



g. typical examples of such verifications.
g. g. g. g.




3. I y1 = cos an y2 = sin 2x ,
g. g. g. g. g. g. y1 =− 2sin g. g. y2 = 2cos 2x , so g. g. g. g. g.



f 2x g. d g. then 2x
g.




y1 = −4 cos 2x = −4
g. g. g. g. g. g. y2 = −4 sin 2x = −4 y2 . y1+ 4y1 = an y2 + 4y2 = 0 .
g. g. g. g. g. g. g. g. g. g. g. g. g. g. g. g. g.




y1 and
g. g. g. Thus 0
g. d



4. I y g . = an y g . = e−3x , y
g. g. g . = 3e3x y g. g . = −3 g. g. y = 9e3x g. g. and
f e3x g. d g. then g . and e−3x , so
g. g. g. g. = 9y g. g.

1 2 1 2 1 1

−3x
2 = 9e
y g. g. g. =2 9 y g. g. g . .


−x −x
y = ex + e−x , y − = (e + e )− (e − e 2
x x
5. I y = ex − e−x , g. g. g. g. g. g. g. g. g. g. g.
g. g. g. g. g. g. g. g.
Thus
f then g. so
g. g. y )=
g. g.
e−x.
g.




y = g . g . y + 2 e−x.
g. g. g.




then
= = y = − 2 e−2x ,
g.
6. I y g . and y g . g. g. g. g.




e−2 x xe −2x ,
f g. g. g.
g.