In Problems 1 and 2, y 1(1 c1ex ) is a one-parameter family of solutions of the first-order DE y y y2 . Find a solution of the first-order IVP consisting of this differential equation and the given initial condition.y(0) 13
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A.I
A.II
APPENDIX I
GAMMA FUNCTION
APPENDIX II
MATRICES
1
Introduction to Differential Equations
1.R
1.1
Definitions and Terminology
1.1
Definitions and Terminology
1.2
Initial-Value Problems
1.2
Initial-Value Problems
1.3
Differential Equations as Mathematical Models
1.3
Differential Equations as Mathematical Models
2
First-order Differential Equations
2.R
2.1
Solution Curves Without a Solution
2.1
Solution Curves Without a Solution
2.2
Separable Equations
2.2
Separable Equations
2.3
Linear Equations
2.3
Linear Equations
2.4
Exact Equations
2.4
Exact Equations
2.5
Solutions by Substitutions
2.5
Solutions by Substitutions
2.6
A Numerical Method
2.6
A Numerical Method
3
Modeling with First-Order Differential Equations
3.R
3.1
Linear Models
3.1
Linear Models
3.2
Nonlinear Models
3.2
Nonlinear Models
3.3
Modeling with Systems of First-Order DEs
3.3
Modeling with Systems of First-Order DEs
4
Higher-Order Differential Equations
4.R
4.1
Preliminary Theory—Linear Equations
4.1
Preliminary Theory—Linear Equations
4.10
Nonlinear Differential Equations
4.2
Reduction of Order
4.2
Reduction of Order
4.3
Homogeneous Linear Equations with Constant Coefficient
4.3
Homogeneous Linear Equations with Constant Coefficient
4.4
Undetermined Coefficients—Superposition Approach
4.4
Undetermined Coefficients—Superposition Approach
4.5
Undetermined Coefficients—Annihilator Approach
4.5
Undetermined Coefficients—Annihilator Approach
4.6
Variation of Parameters
4.6
Variation of Parameters
4.7
Cauchy-Euler Equation
4.7
Cauchy-Euler Equation
4.8
Green’s Functions
4.8
Green’s Functions
4.9
Solving Systems of Linear DEs by Elimination
4.9
Solving Systems of Linear DEs by Elimination
5
Modeling with Higher-Order Differential Equations
5.R
5.1
Linear Models: Initial-Value Problems
5.1
Linear Models: Initial-Value Problems
5.2
Linear Models: Boundary-Value Problems
5.2
Linear Models: Boundary-Value Problems
5.3
Nonlinear Models
5.3
Nonlinear Models
6
Series Solutions of Linear Equations
6.R
6.1
Review of Power Series
6.1
Review of Power Series
6.2
Solutions About Ordinary Points
6.2
Solutions About Ordinary Points
6.3
Solutions About Singular Points
6.3
Solutions About Singular Points
6.4
Special Functions
6.4
Special Functions
7
The Laplace Transform
7.R
7.1
Definition of the Laplace Transform
7.1
Definition of the Laplace Transform
7.2
Inverse Transforms and Transforms of Derivatives
7.2
Inverse Transforms and Transforms of Derivatives
7.3
Operational Properties I
7.3
Operational Properties I
7.4
Operational Properties II
7.4
Operational Properties II
7.5
The Dirac Delta Function
7.5
The Dirac Delta Function
7.6
Systems of Linear Differential Equations
7.6
Systems of Linear Differential Equations
8
Systems of Linear First-Order Differential Equations
8.R
8.1
Preliminary Theory—Linear Systems
8.1
Preliminary Theory—Linear Systems
8.2
Homogeneous Linear Systems
8.2
Homogeneous Linear Systems
8.3
Nonhomogeneous Linear Systems
8.3
Nonhomogeneous Linear Systems
8.4
Matrix Exponential
8.4
Matrix Exponential
9
Numerical Solutions of Ordinary Differential Equations
9.R
9.1
Euler Methods and Error Analysis
9.1
Euler Methods and Error Analysis
9.2
Runge-Kutta Methods
9.2
Runge-Kutta Methods
9.3
Multistep Methods
9.3
Multistep Methods
9.4
Higher-Order Equations and Systems
9.4
Higher-Order Equations and Systems
9.5
Second-Order Boundary-Value Problems
9.5
Second-Order Boundary-Value Problems
Textbook Solutions for A First Course in Differential Equations with Modeling Applications
Chapter 1.2 Problem 29
Question
(a) By inspection find a one-parameter family of solutions of the differential equation xy y. Verify that each member of the family is a solution of the initial-value problem xy y, y(0) 0. (b) Explain part (a) by determining a region R in the xy-plane for which the differential equation xy y would have a unique solution through a point (x0, y0) in R. y
Solution
The first step in solving 1.2 problem number 29 trying to solve the problem we have to refer to the textbook question: (a) By inspection find a one-parameter family of solutions of the differential equation xy y. Verify that each member of the family is a solution of the initial-value problem xy y, y(0) 0. (b) Explain part (a) by determining a region R in the xy-plane for which the differential equation xy y would have a unique solution through a point (x0, y0) in R. y
From the textbook chapter Initial-Value Problems you will find a few key concepts needed to solve this.
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Title
A First Course in Differential Equations with Modeling Applications 10
Author
Dennis G. Zill
ISBN
9781111827052
(a) By inspection find a one-parameter family of solutions of the differential equation
Chapter 1.2 textbook questions
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Chapter 1: Problem 1 A First Course in Differential Equations with Modeling Applications 10
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Chapter 1: Problem 2 A First Course in Differential Equations with Modeling Applications 10
In Problems 1 and 2, y 1(1 c1ex ) is a one-parameter family of solutions of the first-order DE y y y2 . Find a solution of the first-order IVP consisting of this differential equation and the given initial condition.2. y(1) 2
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Chapter 1: Problem 3 A First Course in Differential Equations with Modeling Applications 10
In Problems 36, y 1(x2 c) is a one-parameter family of solutions of the first-order DE y 2xy2 0. Find a y(0) 1 3 solution of the first-order IVP consisting of this differential equation and the given initial condition. Give the largest interval I over which the solution is defined 3
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Chapter 1: Problem 4 A First Course in Differential Equations with Modeling Applications 10
In Problems 36, y 1(x2 c) is a one-parameter family of solutions of the first-order DE y 2xy2 0. Find a y(0) 1 3 solution of the first-order IVP consisting of this differential equation and the given initial condition. Give the largest interval I over which the solution is defined 4
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Chapter 1: Problem 5 A First Course in Differential Equations with Modeling Applications 10
In Problems 36, y 1(x2 c) is a one-parameter family of solutions of the first-order DE y 2xy2 0. Find a y(0) 1 3 solution of the first-order IVP consisting of this differential equation and the given initial condition. Give the largest interval I over which the solution is defined 5
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Chapter 1: Problem 6 A First Course in Differential Equations with Modeling Applications 10
In Problems 36, y 1(x2 c) is a one-parameter family of solutions of the first-order DE y 2xy2 0. Find a y(0) 1 3 solution of the first-order IVP consisting of this differential equation and the given initial condition. Give the largest interval I over which the solution is defined 6
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Chapter 1: Problem 7 A First Course in Differential Equations with Modeling Applications 10
In Problems 710, x c1 cos t c2 sin t is a two-parameter family of solutions of the second-order DE x x 0. Find y(1 2) 4 y(2) 1 2 y(2) 1 3 Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial rea solution of the second-order IVP consisting of this differentialequation and the given initial conditions.
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Chapter 1: Problem 8 A First Course in Differential Equations with Modeling Applications 10
In Problems 710, x c1 cos t c2 sin t is a two-parameter family of solutions of the second-order DE x x 0. Find y(1 2) 4 y(2) 1 2 y(2) 1 3 Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial rea solution of the second-order IVP consisting of this differentialequation and the given initial conditions.
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Chapter 1: Problem 9 A First Course in Differential Equations with Modeling Applications 10
In Problems 710, x c1 cos t c2 sin t is a two-parameter family of solutions of the second-order DE x x 0. Find y(1 2) 4 y(2) 1 2 y(2) 1 3 Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial rea solution of the second-order IVP consisting of this differentialequation and the given initial conditions.
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Chapter 1: Problem 10 A First Course in Differential Equations with Modeling Applications 10
In Problems 710, x c1 cos t c2 sin t is a two-parameter family of solutions of the second-order DE x x 0. Find y(1 2) 4 y(2) 1 2 y(2) 1 3 Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial rea solution of the second-order IVP consisting of this differentialequation and the given initial conditions.
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Chapter 1: Problem 11 A First Course in Differential Equations with Modeling Applications 10
In Problems 1114, y c1ex c2ex is a two-parameter family of solutions of the second-order DE y y 0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions.
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Chapter 1: Problem 12 A First Course in Differential Equations with Modeling Applications 10
In Problems 1114, y c1ex c2ex is a two-parameter family of solutions of the second-order DE y y 0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions.
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Chapter 1: Problem 13 A First Course in Differential Equations with Modeling Applications 10
In Problems 1114, y c1ex c2ex is a two-parameter family of solutions of the second-order DE y y 0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions.
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Chapter 1: Problem 14 A First Course in Differential Equations with Modeling Applications 10
In Problems 1114, y c1ex c2ex is a two-parameter family of solutions of the second-order DE y y 0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions.
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Chapter 1: Problem 15 A First Course in Differential Equations with Modeling Applications 10
In Problems 15 and 16 determine by inspection at least two solutions of the given first-order IV .
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Chapter 1: Problem 16 A First Course in Differential Equations with Modeling Applications 10
In Problems 15 and 16 determine by inspection at least two solutions of the given first-order IV .
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Chapter 1: Problem 17 A First Course in Differential Equations with Modeling Applications 10
In Problems 1724 determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0) in the region.
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Chapter 1: Problem 18 A First Course in Differential Equations with Modeling Applications 10
In Problems 1724 determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0) in the region.
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Chapter 1: Problem 19 A First Course in Differential Equations with Modeling Applications 10
In Problems 1724 determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0) in the region.
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Chapter 1: Problem 20 A First Course in Differential Equations with Modeling Applications 10
In Problems 1724 determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0) in the region.
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Chapter 1: Problem 21 A First Course in Differential Equations with Modeling Applications 10
In Problems 1724 determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0) in the region.
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Chapter 1: Problem 22 A First Course in Differential Equations with Modeling Applications 10
In Problems 1724 determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0) in the region.
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Chapter 1: Problem 23 A First Course in Differential Equations with Modeling Applications 10
In Problems 1724 determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0) in the region.
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Chapter 1: Problem 24 A First Course in Differential Equations with Modeling Applications 10
In Problems 1724 determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0) in the region.
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Chapter 1: Problem 25 A First Course in Differential Equations with Modeling Applications 10
In Problems 2528 determine whether Theorem 1.2.1 guarantees that the differential equation possesses a unique solution through the given point.(1, 4)
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Chapter 1: Problem 26 A First Course in Differential Equations with Modeling Applications 10
In Problems 2528 determine whether Theorem 1.2.1 guarantees that the differential equation possesses a unique solution through the given point.(5, 3)
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Chapter 1: Problem 27 A First Course in Differential Equations with Modeling Applications 10
In Problems 2528 determine whether Theorem 1.2.1 guarantees that the differential equation possesses a unique solution through the given point.. (2, 3)
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Chapter 1: Problem 28 A First Course in Differential Equations with Modeling Applications 10
In Problems 2528 determine whether Theorem 1.2.1 guarantees that the differential equation possesses a unique solution through the given point.. (1, 1)
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Chapter 1: Problem 29 A First Course in Differential Equations with Modeling Applications 10
(a) By inspection find a one-parameter family of solutions of the differential equation xy y. Verify that each member of the family is a solution of the initial-value problem xy y, y(0) 0. (b) Explain part (a) by determining a region R in the xy-plane for which the differential equation xy y would have a unique solution through a point (x0, y0) in R. y
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Chapter 1: Problem 30 A First Course in Differential Equations with Modeling Applications 10
a) Verify that y tan (x c) is a one-parameter family of solutions of the differential equation y 1 y2 . (b) Since f(x, y) 1 y2 and fy 2y are continuous everywhere, the region R in Theorem 1.2.1 can be taken to be the entire xy-plane. Use the family of solutions in part (a) to find an explicit solution of the first-order initial-value problem y 1 y2 , y(0) 0. Even though x0 0 is in the interval (2, 2), explain why the solution is not defined on this interval. (c) Determine the largest interval I of definition for the solution of the initial-value problem in part (b). 31
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Chapter 1: Problem 31 A First Course in Differential Equations with Modeling Applications 10
(a) Verify that y 1(x c) is a one-parameter family of solutions of the differential equation y y2 . (b) Since f(x, y) y2 and fy 2y are continuous everywhere, the region R in Theorem 1.2.1 can be taken to be the entire xy-plane. Find a solution from the family in part (a) that satisfies y(0) 1. Then find a solution from the family in part (a) that satisfies y(0) 1. Determine the largest interval I of definition for the solution of each initial-value problem. (c) Determine the largest interval I of definition for the solution of the first-order initial-value problem y y2 , y(0) 0. [Hint: The solution is not a member of the family of solutions in part (a).] 32.
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Chapter 1: Problem 32 A First Course in Differential Equations with Modeling Applications 10
(a) Show that a solution from the family in part (a) of Problem 31 that satisfies y y2 , y(1) 1, is y 1(2 x). (b) Then show that a solution from the family in part (a) of Problem 31 that satisfies y y2 , y(3) 1, is y 1(2 x). (c) Are the solutions in parts (a) and (b) the same? 33.
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Chapter 1: Problem 33 A First Course in Differential Equations with Modeling Applications 10
(a) Verify that 3x2 y2 c is a one-parameter family of solutions of the differential equation y dydx 3x. (b) By hand, sketch the graph of the implicit solution 3x2 y2 3. Find all explicit solutions y (x) of the DE in part (a) defined by this relation. Give the interval I of definition of each explicit solution (c) The point (2, 3) is on the graph of 3x2 y2 3, but which of the explicit solutions in part (b) satisfies y(2) 3? 34
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Chapter 1: Problem 34 A First Course in Differential Equations with Modeling Applications 10
(a) Use the family of solutions in part (a) of Problem 33 to find an implicit solution of the initial-value(a) Use the family of solutions in part (a) of Problem 33to find an implicit solution of the initial-value
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Chapter 1: Problem 35 A First Course in Differential Equations with Modeling Applications 10
In Problems 3538 the graph of a member of a family of solutions of a second-order differential equation d2 ydx2 f(x, y, y) is given. Match the solution curve with at least one pair of the following initial conditions. (
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Chapter 1: Problem 36 A First Course in Differential Equations with Modeling Applications 10
In Problems 3538 the graph of a member of a family of solutions of a second-order differential equation d2 ydx2 f(x, y, y) is given. Match the solution curve with at least one pair of the following initial conditions. (
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Chapter 1: Problem 37 A First Course in Differential Equations with Modeling Applications 10
In Problems 3538 the graph of a member of a family of solutions of a second-order differential equation d2 ydx2 f(x, y, y) is given. Match the solution curve with at least one pair of the following initial conditions. (
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Chapter 1: Problem 38 A First Course in Differential Equations with Modeling Applications 10
In Problems 3538 the graph of a member of a family of solutions of a second-order differential equation d2 ydx2 f(x, y, y) is given. Match the solution curve with at least one pair of the following initial conditions. (
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Chapter 1: Problem 39 A First Course in Differential Equations with Modeling Applications 10
In Problems 3944, is a twoparameter family of solutions of the second-order DE If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions.
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Chapter 1: Problem 40 A First Course in Differential Equations with Modeling Applications 10
In Problems 3944, is a twoparameter family of solutions of the second-order DE If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions.
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Chapter 1: Problem 41 A First Course in Differential Equations with Modeling Applications 10
In Problems 3944, is a twoparameter family of solutions of the second-order DE If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions.
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Chapter 1: Problem 42 A First Course in Differential Equations with Modeling Applications 10
In Problems 3944, is a twoparameter family of solutions of the second-order DE If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions.
Read more -
Chapter 1: Problem 43 A First Course in Differential Equations with Modeling Applications 10
In Problems 3944, is a twoparameter family of solutions of the second-order DE If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions.
Read more -
Chapter 1: Problem 44 A First Course in Differential Equations with Modeling Applications 10
In Problems 3944, is a twoparameter family of solutions of the second-order DE If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions.
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Chapter 1: Problem 45 A First Course in Differential Equations with Modeling Applications 10
Find a function y f(x) whose graph at each point (x, y) has the slope given by 8e2x 6x and has the y-intercept (0, 9)
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Chapter 1: Problem 46 A First Course in Differential Equations with Modeling Applications 10
Find a function y f(x) whose second derivative is y 12x 2 at each point (x, y) on its graph and y x 5 is tangent to the graph at the point corresponding to x 1.
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Chapter 1: Problem 47 A First Course in Differential Equations with Modeling Applications 10
Consider the initial-value problem y x 2y, . Determine which of the two curves shown in Figure 1.2.11 is the only plausible solution curve. Explain your reasoning.
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Chapter 1: Problem 48 A First Course in Differential Equations with Modeling Applications 10
Determine a plausible value of x0 for which the graph of the solution of the initial-value problem y 2y 3x 6, y(x0) 0 is tangent to the x-axis at (x0, 0). Explain your reasoning.
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Chapter 1: Problem 49 A First Course in Differential Equations with Modeling Applications 10
Suppose that the first-order differential equation dydx f(x, y) possesses a one-parameter family of solutions and that f(x, y) satisfies the hypotheses of Theorem 1.2.1 in some rectangular region R of the xy-plane. Explain why two different solution curves cannot intersect or be tangent to each other at a point (x0, y0) in R.
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Chapter 1: Problem 50 A First Course in Differential Equations with Modeling Applications 10
The functions and have the same domain but are clearly different. See Figures 1.2.12(a) and 1.2.12(b), respectively. Show that both functions are solutions of the initial-value problem dydx xy1/2, y(2) 1 on the interval (, ). Resolve the apparent contradiction between this fact and the last sentence in Example 5.
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Chapter 1: Problem 51 A First Course in Differential Equations with Modeling Applications 10
Beginning in the next section we will see that differential equations can be used to describe or model many different physical systems. In this problem suppose that a model of the growing population of a small community is given by the initial-value problem where P is the number of individuals in the community and time t is measured in years. How fastthat is, at what rateis the population increasing at t 0? How fast is the population increasing when the population is 500?
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