In each of Problems 1 through 14: (a) Seek power series solutions of the given differential equation about the given point x0; find the recurrence relation. (b) Find the first four terms in each of two solutions y1 and y2 (unless the series terminates sooner). (c) By evaluating the Wronskian W(y1, y2)(x0), show that y1 and y2 form a fundamental set of solutions. (d) If possible, find the general term in each solution.y y = 0, x0 = 0
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Table of Contents
1.1
Some Basic Mathematical Models; Direction Fields
1.2
Solutions of Some Differential Equations
1.3
Classification of Differential Equations
2
First Order Difference Equations
2.1
Linear Equations; Method of Integrating Factors
2.2
Separable Equations
2.3
Modeling with First Order Equations
2.4
Differences Between Linear and Nonlinear Equations
2.5
Autonomous Equations and Population Dynamics
2.6
Exact Equations and Integrating Factors
2.7
Numerical Approximations: Eulers Method
2.8
The Existence and Uniqueness Theorem
2.9
First Order Difference Equations
3.1
Homogeneous Equations with Constant Coefficients
3.2
Solutions of Linear Homogeneous Equations; the Wronskian
3.3
Complex Roots of the Characteristic Equation
3.4
Repeated Roots; Reduction of Order
3.5
Nonhomogeneous Equations; Method of Undetermined Coefficients
3.6
Variation of Parameters
3.7
Mechanical and Electrical Vibrations
3.8
Forced Vibrations
4.1
General Theory of nth Order Linear Equations
4.2
Homogeneous Equations with Constant Coefficients
4.3
The Method of Undetermined Coefficients
4.4
The Method of Variation of Parameters
5.1
Review of Power Series
5.2
Series Solutions Near an Ordinary Point, Part I
5.3
Series Solutions Near an Ordinary Point, Part II
5.4
Euler Equations; Regular Singular Points
5.5
Series Solutions Near a Regular Singular Point, Part I
5.6
Series Solutions Near a Regular Singular Point, Part II
5.7
Bessels Equation
6.1
Definition of the Laplace Transform
6.2
Solution of Initial Value Problems
6.3
Step Functions
6.4
Differential Equations with Discontinuous Forcing Functions
6.5
Impulse Functions
6.6
The Convolution Integral
7.1
Introduction
7.2
Review of Matrices
7.3
Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors
7.4
Basic Theory of Systems of First Order Linear Equations
7.5
Homogeneous Linear Systems with Constant Coefficients
7.6
Complex Eigenvalues
7.7
Fundamental Matrices
7.8
Repeated Eigenvalues
7.9
Nonhomogeneous Linear Systems
8.1
The Euler or Tangent Line Method
8.2
Improvements on the Euler Method
8.3
The RungeKutta Method
8.4
Multistep Methods
8.5
More on Errors; Stability
8.6
Systems of First Order Equations
9.1
The Phase Plane: Linear Systems
9.2
Autonomous Systems and Stability
9.3
Locally Linear Systems
9.4
Competing Species
9.5
PredatorPrey Equations
9.6
Liapunovs Second Method
9.7
Periodic Solutions and Limit Cycles
9.8
Chaos and Strange Attractors: The Lorenz Equations
10.1
Two-Point Boundary Value Problems
10.2
Fourier Series
10.3
The Fourier Convergence Theorem
10.4
Even and Odd Functions
10.5
Separation of Variables; Heat Conduction in a Rod
10.6
Other Heat Conduction Problems
10.7
The Wave Equation: Vibrations of an Elastic String
10.8
Laplaces Equation
11.1
The Occurrence of Two-Point Boundary Value Problems
11.2
SturmLiouville Boundary Value Problems
11.3
Nonhomogeneous Boundary Value Problems
11.4
Singular SturmLiouville Problems
11.5
Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion
11.6
Series of Orthogonal Functions: Mean Convergence
Textbook Solutions for Elementary Differential Equations and Boundary Value Problems
Chapter 5.2 Problem 6
Question
In each of 1 through 14: (a) Seek power series solutions of the given differential equation about the given point x0; find the recurrence relation. (b) Find the first four terms in each of two solutions y1 and y2 (unless the series terminates sooner). (c) By evaluating the Wronskian W(y1, y2)(x0), show that y1 and y2 form a fundamental set of solutions. (d) If possible, find the general term in each solution.. (2 + x2)y xy + 4y = 0, x0 = 0
Solution
The first step in solving 5.2 problem number 6 trying to solve the problem we have to refer to the textbook question: In each of 1 through 14: (a) Seek power series solutions of the given differential equation about the given point x0; find the recurrence relation. (b) Find the first four terms in each of two solutions y1 and y2 (unless the series terminates sooner). (c) By evaluating the Wronskian W(y1, y2)(x0), show that y1 and y2 form a fundamental set of solutions. (d) If possible, find the general term in each solution.. (2 + x2)y xy + 4y = 0, x0 = 0
From the textbook chapter Series Solutions Near an Ordinary Point, Part I you will find a few key concepts needed to solve this.
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Title
Elementary Differential Equations and Boundary Value Problems 9
Author
Boyce, Richard C. DiPrima
ISBN
9780470383346
See answer: In each of 1 through 14: (a) Seek power series solutions of the given
Chapter 5.2 textbook questions
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Chapter 5: Problem 1 Elementary Differential Equations and Boundary Value Problems 9
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Chapter 5: Problem 2 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 14: (a) Seek power series solutions of the given differential equation about the given point x0; find the recurrence relation. (b) Find the first four terms in each of two solutions y1 and y2 (unless the series terminates sooner). (c) By evaluating the Wronskian W(y1, y2)(x0), show that y1 and y2 form a fundamental set of solutions. (d) If possible, find the general term in each solution. xy y = 0, x0 = 0
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Chapter 5: Problem 3 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 14: (a) Seek power series solutions of the given differential equation about the given point x0; find the recurrence relation. (b) Find the first four terms in each of two solutions y1 and y2 (unless the series terminates sooner). (c) By evaluating the Wronskian W(y1, y2)(x0), show that y1 and y2 form a fundamental set of solutions. (d) If possible, find the general term in each solution.
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Chapter 5: Problem 4 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 14: (a) Seek power series solutions of the given differential equation about the given point x0; find the recurrence relation. (b) Find the first four terms in each of two solutions y1 and y2 (unless the series terminates sooner). (c) By evaluating the Wronskian W(y1, y2)(x0), show that y1 and y2 form a fundamental set of solutions. (d) If possible, find the general term in each solution.. y + k2x2y = 0, x0 = 0, k a constant
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Chapter 5: Problem 5 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 14: (a) Seek power series solutions of the given differential equation about the given point x0; find the recurrence relation. (b) Find the first four terms in each of two solutions y1 and y2 (unless the series terminates sooner). (c) By evaluating the Wronskian W(y1, y2)(x0), show that y1 and y2 form a fundamental set of solutions. (d) If possible, find the general term in each solution.(1 x)y + y = 0, x0 = 0
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Chapter 5: Problem 6 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 14: (a) Seek power series solutions of the given differential equation about the given point x0; find the recurrence relation. (b) Find the first four terms in each of two solutions y1 and y2 (unless the series terminates sooner). (c) By evaluating the Wronskian W(y1, y2)(x0), show that y1 and y2 form a fundamental set of solutions. (d) If possible, find the general term in each solution.. (2 + x2)y xy + 4y = 0, x0 = 0
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Chapter 5: Problem 7 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 14: (a) Seek power series solutions of the given differential equation about the given point x0; find the recurrence relation. (b) Find the first four terms in each of two solutions y1 and y2 (unless the series terminates sooner). (c) By evaluating the Wronskian W(y1, y2)(x0), show that y1 and y2 form a fundamental set of solutions. (d) If possible, find the general term in each solution.
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Chapter 5: Problem 8 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 14: (a) Seek power series solutions of the given differential equation about the given point x0; find the recurrence relation. (b) Find the first four terms in each of two solutions y1 and y2 (unless the series terminates sooner). (c) By evaluating the Wronskian W(y1, y2)(x0), show that y1 and y2 form a fundamental set of solutions. (d) If possible, find the general term in each solution.xy + y + xy = 0, x0 = 1
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Chapter 5: Problem 9 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 14: (a) Seek power series solutions of the given differential equation about the given point x0; find the recurrence relation. (b) Find the first four terms in each of two solutions y1 and y2 (unless the series terminates sooner). (c) By evaluating the Wronskian W(y1, y2)(x0), show that y1 and y2 form a fundamental set of solutions. (d) If possible, find the general term in each solution.
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Chapter 5: Problem 10 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 14: (a) Seek power series solutions of the given differential equation about the given point x0; find the recurrence relation. (b) Find the first four terms in each of two solutions y1 and y2 (unless the series terminates sooner). (c) By evaluating the Wronskian W(y1, y2)(x0), show that y1 and y2 form a fundamental set of solutions. (d) If possible, find the general term in each solution.(4 x2)y + 2y = 0, x0 = 0
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Chapter 5: Problem 11 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 14: (a) Seek power series solutions of the given differential equation about the given point x0; find the recurrence relation. (b) Find the first four terms in each of two solutions y1 and y2 (unless the series terminates sooner). (c) By evaluating the Wronskian W(y1, y2)(x0), show that y1 and y2 form a fundamental set of solutions. (d) If possible, find the general term in each solution.. (3 x2)y 3xy y = 0, x0 = 0
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Chapter 5: Problem 12 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 14: (a) Seek power series solutions of the given differential equation about the given point x0; find the recurrence relation. (b) Find the first four terms in each of two solutions y1 and y2 (unless the series terminates sooner). (c) By evaluating the Wronskian W(y1, y2)(x0), show that y1 and y2 form a fundamental set of solutions. (d) If possible, find the general term in each solution.(1 x)y + xy y = 0, x0 = 0
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Chapter 5: Problem 13 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 14: (a) Seek power series solutions of the given differential equation about the given point x0; find the recurrence relation. (b) Find the first four terms in each of two solutions y1 and y2 (unless the series terminates sooner). (c) By evaluating the Wronskian W(y1, y2)(x0), show that y1 and y2 form a fundamental set of solutions. (d) If possible, find the general term in each solution.
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Chapter 5: Problem 14 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 14: (a) Seek power series solutions of the given differential equation about the given point x0; find the recurrence relation. (b) Find the first four terms in each of two solutions y1 and y2 (unless the series terminates sooner). (c) By evaluating the Wronskian W(y1, y2)(x0), show that y1 and y2 form a fundamental set of solutions. (d) If possible, find the general term in each solution.2y + (x + 1)y + 3y = 0, x0 = 2
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Chapter 5: Problem 15 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 15 through 18: (a) Find the first five nonzero terms in the solution of the given initial value problem. (b) Plot the four-term and the five-term approximations to the solution on the same axes. (c) From the plot in part (b) estimate the interval in which the four-term approximation is reasonably accurate.. y xy y = 0, y(0) = 2, y (0) = 1; see Problem 2
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Chapter 5: Problem 16 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 15 through 18: (a) Find the first five nonzero terms in the solution of the given initial value problem. (b) Plot the four-term and the five-term approximations to the solution on the same axes. (c) From the plot in part (b) estimate the interval in which the four-term approximation is reasonably accurate.
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Chapter 5: Problem 17 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 15 through 18: (a) Find the first five nonzero terms in the solution of the given initial value problem. (b) Plot the four-term and the five-term approximations to the solution on the same axes. (c) From the plot in part (b) estimate the interval in which the four-term approximation is reasonably accurate.y + xy + 2y = 0, y(0) = 4, y (0) = 1; see Problem 7
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Chapter 5: Problem 18 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 15 through 18: (a) Find the first five nonzero terms in the solution of the given initial value problem. (b) Plot the four-term and the five-term approximations to the solution on the same axes. (c) From the plot in part (b) estimate the interval in which the four-term approximation is reasonably accurate.(1 x)y + xy y = 0, y(0) = 3, y (0) = 2; see Problem 12
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Chapter 5: Problem 19 Elementary Differential Equations and Boundary Value Problems 9
(a) By making the change of variable x 1 = t and assuming that y has a Taylor series in powers of t, find two series solutions of y + (x 1) 2 y + (x2 1)y = 0 in powers of x 1. (b) Show that you obtain the same result by assuming that y has a Taylor series in powers of x 1 and also expressing the coefficient x2 1 in powers of x 1.
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Chapter 5: Problem 20 Elementary Differential Equations and Boundary Value Problems 9
Show directly, using the ratio test, that the two series solutions of Airys equation about x = 0 converge for all x; see Eq. (20) of the text
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Chapter 5: Problem 21 Elementary Differential Equations and Boundary Value Problems 9
The Hermite Equation. The equation y 2xy + y = 0, < x < , where is a constant, is known as the Hermite5 equation. It is an important equation in mathematical physics. (a) Find the first four terms in each of two solutions about x = 0 and show that they form a fundamental set of solutions. (b) Observe that if is a nonnegative even integer, then one or the other of the series solutions terminates and becomes a polynomial. Find the polynomial solutions for = 0, 2, 4, 6, 8, and 10. Note that each polynomial is determined only up to a multiplicative constant. (c) The Hermite polynomial Hn(x) is defined as the polynomial solution of the Hermite equation with = 2n for which the coefficient of xn is 2n. Find H0(x), ... , H5(x).
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Chapter 5: Problem 22 Elementary Differential Equations and Boundary Value Problems 9
Consider the initial value problem y = 1 y2, y(0) = 0. (a) Show that y = sin x is the solution of this initial value problem. (b) Look for a solution of the initial value problem in the form of a power series about x = 0. Find the coefficients up to the term in x3 in this series.
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Chapter 5: Problem 23 Elementary Differential Equations and Boundary Value Problems 9
Consider the initial value problem y = 1 y2, y(0) = 0. (a) Show that y = sin x is the solution of this initial value problem. (b) Look for a solution of the initial value problem in the form of a power series about x = 0. Find the coefficients up to the term in x3 in this series.y xy y = 0, y(0) = 1, y (0) = 0; see Problem 2
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Chapter 5: Problem 24 Elementary Differential Equations and Boundary Value Problems 9
Consider the initial value problem y = 1 y2, y(0) = 0. (a) Show that y = sin x is the solution of this initial value problem. (b) Look for a solution of the initial value problem in the form of a power series about x = 0. Find the coefficients up to the term in x3 in this series.
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Chapter 5: Problem 25 Elementary Differential Equations and Boundary Value Problems 9
Consider the initial value problem y = 1 y2, y(0) = 0. (a) Show that y = sin x is the solution of this initial value problem. (b) Look for a solution of the initial value problem in the form of a power series about x = 0. Find the coefficients up to the term in x3 in this series.y + xy + 2y = 0, y(0) = 0, y (0) = 1; see Problem 7
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Chapter 5: Problem 26 Elementary Differential Equations and Boundary Value Problems 9
Consider the initial value problem y = 1 y2, y(0) = 0. (a) Show that y = sin x is the solution of this initial value problem. (b) Look for a solution of the initial value problem in the form of a power series about x = 0. Find the coefficients up to the term in x3 in this series.. (4 x2)y + 2y = 0, y(0) = 0, y (0) = 1; see Problem 10
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Chapter 5: Problem 27 Elementary Differential Equations and Boundary Value Problems 9
Consider the initial value problem y = 1 y2, y(0) = 0. (a) Show that y = sin x is the solution of this initial value problem. (b) Look for a solution of the initial value problem in the form of a power series about x = 0. Find the coefficients up to the term in x3 in this series.y + x2y = 0, y(0) = 1, y (0) = 0; see Problem 4
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Chapter 5: Problem 28 Elementary Differential Equations and Boundary Value Problems 9
Consider the initial value problem y = 1 y2, y(0) = 0. (a) Show that y = sin x is the solution of this initial value problem. (b) Look for a solution of the initial value problem in the form of a power series about x = 0. Find the coefficients up to the term in x3 in this series.(1 x)y + xy 2y = 0, y(0) = 0, y (0) = 1
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