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.
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Table of Contents
1
INTRODUCTION TO DIFFERENTIAL EQUATIONS
1.1
Definitions and Terminology
1.2
Initial-Value Problems
1.3
Differential Equations as Mathematical Models
2
FIRST-ORDER DIFFERENTIAL EQUATIONS
2.1
Solution Curves Without a Solution
2.2
Separable Variables
2.3
Linear Equations
2.4
Exact Equations
2.5
Solutions by Substitutions
2.6
A Numerical Method
3
MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS
3.1
Linear Models
3.2
Nonlinear Models
3.3
Modeling with Systems of First-Order DEs
4
HIGHER-ORDER DIFFERENTIAL EQUATIONS
4.1
Preliminary TheoryLinear Equations
4.2
Reduction of Order
4.3
Homogeneous Linear Equations with Constant Coefficients
4.4
Undetermined CoefficientsSuperposition Approach
4.5
Undetermined CoefficientsAnnihilator Approach
4.6
Variation of Parameters
4.7
Cauchy-Euler Equation
4.8
Solving Systems of Linear DEs by Elimination
4.9
Nonlinear Differential Equations
5
MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS
5.1
Linear Models: Initial-Value Problems
5.2
Linear Models: Boundary-Value Problems
5.3
Nonlinear Models
6
SERIES SOLUTIONS OF LINEAR EQUATIONS
6.1
Solutions About Ordinary Points
6.2
Solutions About Singular Points
6.3
Special Functions
7
THE LAPLACE TRANSFORM
7.1
Definition of the Laplace Transform
7.2
Inverse Transforms and Transforms of Derivatives
7.3
Operational Properties I
7.4
Operational Properties II
7.5
The Dirac Delta Function
7.6
Systems of Linear Differential Equations
8
SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS
8.1
Preliminary TheoryLinear Systems
8.2
Homogeneous Linear Systems
8.3
Nonhomogeneous Linear Systems
8.4
Matrix Exponential
9
NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS
9.1
Euler Methods and Error Analysis
9.2
Runge-Kutta Methods
9.3
Multistep Methods
9.4
Higher-Order Equations and Systems
9.5
Second-Order Boundary-Value Problems
10
PLANE AUTONOMOUS SYSTEMS
10.1
Autonomous Systems
10.2
Stability of Linear Systems
10.3
Linearization and Local Stability
10.4
Autonomous Systems as Mathematical Models
11
ORTHOGONAL FUNCTIONS AND FOURIER SERIES
11.1
Orthogonal Functions
11.2
Fourier Series
11.3
Fourier Cosine and Sine Series
11.4
Sturm-Liouville Problem
11.5
Bessel and Legendre Series
12
BOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES
12.1
Separable Partial Differential Equations
12.2
Classical PDEs and Boundary-Value Problems
12.3
Heat Equation
12.4
Wave Equation
12.5
Laplaces Equation
12.6
Nonhomogeneous Boundary-Value Problems
12.7
Orthogonal Series Expansions
12.8
Higher-Dimensional Problems
13
BOUNDARY-VALUE PROBLEMS IN OTHER COORDINATE SYSTEMS
13.1
Polar Coordinates
13.2
Polar and Cylindrical Coordinates
13.3
Spherical Coordinates
14
INTEGRAL TRANSFORMS
14.1
Error Function
14.2
Laplace Transform
14.3
Fourier Integral
14.4
Fourier Transforms
15
NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
15.1
Laplaces Equation
15.2
Heat Equation
15.3
Wave Equation
Textbook Solutions for Differential Equations with Boundary-Value Problems
Chapter 1.2 Problem 3
Question
In 36, y 1(x2 c) is a one-parameter family of solutions of the first-order DE y 2xy2 0. Find a 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
Solution
The first step in solving 1.2 problem number 3 trying to solve the problem we have to refer to the textbook question: In 36, y 1(x2 c) is a one-parameter family of solutions of the first-order DE y 2xy2 0. Find a 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
From the textbook chapter Initial-Value Problems you will find a few key concepts needed to solve this.
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Title
Differential Equations with Boundary-Value Problems 7
Author
Dennis G. Zill, Michael R. Cullen
ISBN
9780495108368
Solution: In 36, y 1(x2 c) is a one-parameter family of solutions of the first-order DE
Chapter 1.2 textbook questions
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Chapter 1: Problem 1 Differential Equations with Boundary-Value Problems 7
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Chapter 1: Problem 2 Differential Equations with Boundary-Value Problems 7
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.
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Chapter 1: Problem 3 Differential Equations with Boundary-Value Problems 7
In Problems 36, y 1(x2 c) is a one-parameter family of solutions of the first-order DE y 2xy2 0. Find a 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 Differential Equations with Boundary-Value Problems 7
In Problems 36, y 1(x2 c) is a one-parameter family of solutions of the first-order DE y 2xy2 0. Find a 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 5 Differential Equations with Boundary-Value Problems 7
In Problems 36, y 1(x2 c) is a one-parameter family of solutions of the first-order DE y 2xy2 0. Find a 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 6 Differential Equations with Boundary-Value Problems 7
In Problems 36, y 1(x2 c) is a one-parameter family of solutions of the first-order DE y 2xy2 0. Find a 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 7 Differential Equations with Boundary-Value Problems 7
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 a solution of the second-order IVP consisting of this differential equation and the given initial conditions. 7
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Chapter 1: Problem 8 Differential Equations with Boundary-Value Problems 7
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 a solution of the second-order IVP consisting of this differential equation and the given initial conditions. 7
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Chapter 1: Problem 9 Differential Equations with Boundary-Value Problems 7
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 a solution of the second-order IVP consisting of this differential equation and the given initial conditions. 7
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Chapter 1: Problem 10 Differential Equations with Boundary-Value Problems 7
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 a solution of the second-order IVP consisting of this differential equation and the given initial conditions. 7
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Chapter 1: Problem 11 Differential Equations with Boundary-Value Problems 7
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 Differential Equations with Boundary-Value Problems 7
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 Differential Equations with Boundary-Value Problems 7
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 Differential Equations with Boundary-Value Problems 7
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 Differential Equations with Boundary-Value Problems 7
In Problems 15 and 16 determine by inspection at least two solutions of the given first-order IVP.
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Chapter 1: Problem 16 Differential Equations with Boundary-Value Problems 7
In Problems 15 and 16 determine by inspection at least two solutions of the given first-order IVP.
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Chapter 1: Problem 17 Differential Equations with Boundary-Value Problems 7
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 Differential Equations with Boundary-Value Problems 7
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 Differential Equations with Boundary-Value Problems 7
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 Differential Equations with Boundary-Value Problems 7
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 Differential Equations with Boundary-Value Problems 7
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 Differential Equations with Boundary-Value Problems 7
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 Differential Equations with Boundary-Value Problems 7
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 Differential Equations with Boundary-Value Problems 7
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 Differential Equations with Boundary-Value Problems 7
In Problems 2528 determine whether Theorem 1.2.1 guarantees that the differential equation possesses a unique solution through the given point.
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Chapter 1: Problem 26 Differential Equations with Boundary-Value Problems 7
In Problems 2528 determine whether Theorem 1.2.1 guarantees that the differential equation possesses a unique solution through the given point.
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Chapter 1: Problem 27 Differential Equations with Boundary-Value Problems 7
In Problems 2528 determine whether Theorem 1.2.1 guarantees that the differential equation possesses a unique solution through the given point.
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Chapter 1: Problem 28 Differential Equations with Boundary-Value Problems 7
In Problems 2528 determine whether Theorem 1.2.1 guarantees that the differential equation possesses a unique solution through the given point.
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Chapter 1: Problem 29 Differential Equations with Boundary-Value Problems 7
(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. (c) Verify that the piecewise-defined function satisfies the condition y(0) 0. Determine whether this function is also a solution of the initial-value problem in part (a).
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Chapter 1: Problem 30 Differential Equations with Boundary-Value Problems 7
(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 f y 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 Differential Equations with Boundary-Value Problems 7
(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 f y 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).]
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Chapter 1: Problem 32 Differential Equations with Boundary-Value Problems 7
(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?
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Chapter 1: Problem 33 Differential Equations with Boundary-Value Problems 7
(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?
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Chapter 1: Problem 34 Differential Equations with Boundary-Value Problems 7
(a) Use the family of solutions in part (a) of Problem 33 to find an implicit solution of the initial-value problem y dydx 3x, y(2) 4. Then, by hand, sketch the graph of the explicit solution of this problem and give its interval I of definition. (b) Are there any explicit solutions of y dydx 3x that pass through the origin?
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Chapter 1: Problem 35 Differential Equations with Boundary-Value Problems 7
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. (a) y(1) 1, y(1) 2 (b) y(1) 0, y(1) 4 (c) y(1) 1, y(1) 2 (d) y(0) 1, y(0) 2 (e) y(0) 1, y(0) 0 (f) y(0) 4, y(0) 2
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Chapter 1: Problem 36 Differential Equations with Boundary-Value Problems 7
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. (a) y(1) 1, y(1) 2 (b) y(1) 0, y(1) 4 (c) y(1) 1, y(1) 2 (d) y(0) 1, y(0) 2 (e) y(0) 1, y(0) 0 (f) y(0) 4, y(0) 2
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Chapter 1: Problem 37 Differential Equations with Boundary-Value Problems 7
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. (a) y(1) 1, y(1) 2 (b) y(1) 0, y(1) 4 (c) y(1) 1, y(1) 2 (d) y(0) 1, y(0) 2 (e) y(0) 1, y(0) 0 (f) y(0) 4, y(0) 2
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Chapter 1: Problem 38 Differential Equations with Boundary-Value Problems 7
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. (a) y(1) 1, y(1) 2 (b) y(1) 0, y(1) 4 (c) y(1) 1, y(1) 2 (d) y(0) 1, y(0) 2 (e) y(0) 1, y(0) 0 (f) y(0) 4, y(0) 2
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Chapter 1: Problem 39 Differential Equations with Boundary-Value Problems 7
In Problems 39 and 40 use Problem 51 in Exercises 1.1 and (2) and (3) of this section.
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Chapter 1: Problem 40 Differential Equations with Boundary-Value Problems 7
In Problems 39 and 40 use Problem 51 in Exercises 1.1 and (2) and (3) of this section.
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Chapter 1: Problem 41 Differential Equations with Boundary-Value Problems 7
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 42 Differential Equations with Boundary-Value Problems 7
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 43 Differential Equations with Boundary-Value Problems 7
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 44 Differential Equations with Boundary-Value Problems 7
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 45 Differential Equations with Boundary-Value Problems 7
Population Growth 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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