For Problems 15, determine all singular points of the given differential equation and classify them as regular or irregular singular points.y + 1 1 x y + x y = 0. 2
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Table of Contents
1.1
Differential Equations Everywhere
1.10
Numerical Solution to First-Order Differential Equations
1.11
Some Higher-Order Differential Equations
1.12
Basic Theory of Differential Equations
1.2
Basic Ideas and Terminology
1.3
The Geometry of First-Order Differential Equations
1.4
Separable Differential Equations
1.5
Some Simple Population Models
1.6
First-Order Linear Differential Equations
1.7
Modeling Problems Using First-Order Linear Differential Equations
1.8
Change of Variables
1.9
Exact Differential Equations
2.1
Matrices: Definitions and Notation
2.2
Matrix Algebra
2.3
Terminology for Systems of Linear Equations
2.4
Row-Echelon Matrices and Elementary Row Operations
2.5
Gaussian Elimination
2.6
The Inverse of a Square Matrix
2.7
Elementary Matrices and the LU Factorization
2.8
The Invertible Matrix Theorem I I
2.9
Chapter Review
3.1
The Definition of the Determinant
3.2
Properties of Determinants F
3.3
Cofactor Expansions
3.4
Summary of Determinants
3.5
Chapter Review
4
Vector Spaces
4.1
Vectors in Rn
4.2
Definition of a Vector Space
4.3
Subspaces
4.4
Spanning Sets
4.5
Linear Dependence and Linear Independence
4.6
Bases and Dimension
4.7
Change of Basis
4.8
Row Space and Column Space
4.9
The Rank-Nullity Theorem
5.1
Definition of an Inner Product Space
5.2
Orthogonal Sets of Vectors and Orthogonal Projections
5.4
Least Squares Approximation
5.5
Inner Product Spaces
6.1
Definition of a Linear Transformation
6.2
Transformations of R2
6.3
The Kernel and Range of a Linear Transformation
6.4
Additional Properties of Linear Transformations
6.5
The Matrix of a Linear Transformation
6.6
Linear Transformations
7.1
The Eigenvalue/Eigenvector Problem
7.2
General Results for Eigenvalues and Eigenvectors
7.3
Diagonalization
7.4
An Introduction to the Matrix Exponential Function
7.5
Orthogonal Diagonalization and Quadratic Forms
7.6
Jordan Canonical Forms
7.7
The Algebraic Eigenvalue/Eigenvector Problem
8.1
General Theory for Linear Differential Equations
8.10
Linear Differential Equations of Order n
8.2
Constant Coefficient Homogeneous Linear Differential Equations
8.3
The Method of Undetermined Coefficients: Annihilators
8.4
Complex-Valued Trial Solutions
8.5
Oscillations of a Mechanical System
8.6
RLC Circuits
8.7
The Variation of Parameters Method
8.8
A Differential Equation with Nonconstant Coefficients
8.9
Reduction of Order
9.1
First-Order Linear Systems
9.10
Nonlinear Systems
9.11
Systems of Differential Equations
9.2
Vector Formulation
9.3
General Results for First-Order Linear Differential Systems
9.4
Vector Differential Equations: Nondefective Coefficient Matrix
9.5
Vector Differential Equations: Defective Coefficient Matrix
9.6
Variation-of-Parameters for Linear Systems
9.7
Some Applications of Linear Systems of Differential Equations
9.8
Matrix Exponential Function and Systems of Differential Equations
9.9
The Phase Plane for Linear Autonomous Systems
10.1
Definition of the Laplace Transform
10.10
The Laplace Transform and Some Elementary Applications
10.2
The Existence of the Laplace Transform and the Inverse Transform
10.3
Periodic Functions and the Laplace Transform
10.4
The Transform of Derivatives and Solution of Initial-Value Problems
10.5
The First Shifting Theorem
10.6
The Unit Step Function
10.7
The Second Shifting Theorem
10.8
Impulsive Driving Terms: The Dirac Delta Function
10.9
The Convolution Integral
11.1
Review of Power Series
11.2
Series Solutions about an Ordinary Point
11.3
The Legendre Equation
11.4
Series Solutions about a Regular Singular Point
11.5
Frobenius Theory
11.6
Bessels Equation of Order p
11.7
Series Solutions to Linear Differential Equations
Textbook Solutions for Differential Equations and Linear Algebra
Chapter 11.4 Problem 21
Question
Find two linearly independent solutions to x2 y + x(3 2x)y + (1 2x)y = 0 on (0,).
Solution
The first step in solving 11.4 problem number 21 trying to solve the problem we have to refer to the textbook question: Find two linearly independent solutions to x2 y + x(3 2x)y + (1 2x)y = 0 on (0,).
From the textbook chapter Series Solutions about a Regular Singular Point you will find a few key concepts needed to solve this.
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full solution
Title
Differential Equations and Linear Algebra 4
Author
Stephen W. Goode
ISBN
9780321964670
Find two linearly independent solutions to x2 y + x(3 2x)y + (1 2x)y = 0 on (0,)
Chapter 11.4 textbook questions
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Chapter 11: Problem 1 Differential Equations and Linear Algebra 4
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Chapter 11: Problem 2 Differential Equations and Linear Algebra 4
For Problems 15, determine all singular points of the given differential equation and classify them as regular or irregular singular points.x2 y + x x2 4 y + 1 x(x 2)(x + 2)2 y = 0. 3
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Chapter 11: Problem 3 Differential Equations and Linear Algebra 4
For Problems 15, determine all singular points of the given differential equation and classify them as regular or irregular singular points.
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Chapter 11: Problem 4 Differential Equations and Linear Algebra 4
For Problems 15, determine all singular points of the given differential equation and classify them as regular or irregular singular points.(x 2)2 y + (x 2)ex y + 4 x y = 0. 5
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Chapter 11: Problem 5 Differential Equations and Linear Algebra 4
For Problems 15, determine all singular points of the given differential equation and classify them as regular or irregular singular points.y + 2 x(x 3) y 1 x3(x + 3) y = 0. F
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Chapter 11: Problem 6 Differential Equations and Linear Algebra 4
For Problems 69, determine the roots of the indicial equation of the given differential equation.x2 y + x(1 x)y 7y = 0. 7
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Chapter 11: Problem 7 Differential Equations and Linear Algebra 4
For Problems 69, determine the roots of the indicial equation of the given differential equation.
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Chapter 11: Problem 8 Differential Equations and Linear Algebra 4
For Problems 69, determine the roots of the indicial equation of the given differential equation.4x y x y + 2y = 0. 9
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Chapter 11: Problem 9 Differential Equations and Linear Algebra 4
For Problems 69, determine the roots of the indicial equation of the given differential equation.x2 y x(cos x)y + 5e2x y = 0. F
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Chapter 11: Problem 10 Differential Equations and Linear Algebra 4
For Problems 1017, show that the indicial equation of the given differential equation has distinct roots that do not differ by an integer and find two linearly independent Frobenius series solutions on (0,).4x2 y + 3x y + x y = 0. 1
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Chapter 11: Problem 11 Differential Equations and Linear Algebra 4
For Problems 1017, show that the indicial equation of the given differential equation has distinct roots that do not differ by an integer and find two linearly independent Frobenius series solutions on (0,).
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Chapter 11: Problem 12 Differential Equations and Linear Algebra 4
For Problems 1017, show that the indicial equation of the given differential equation has distinct roots that do not differ by an integer and find two linearly independent Frobenius series solutions on (0,).x2 y + x y (2 + x)y = 0. 1
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Chapter 11: Problem 13 Differential Equations and Linear Algebra 4
For Problems 1017, show that the indicial equation of the given differential equation has distinct roots that do not differ by an integer and find two linearly independent Frobenius series solutions on (0,).
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Chapter 11: Problem 14 Differential Equations and Linear Algebra 4
For Problems 1017, show that the indicial equation of the given differential equation has distinct roots that do not differ by an integer and find two linearly independent Frobenius series solutions on (0,).3x2 y x(x + 8)y + 6y = 0. 1
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Chapter 11: Problem 15 Differential Equations and Linear Algebra 4
For Problems 1017, show that the indicial equation of the given differential equation has distinct roots that do not differ by an integer and find two linearly independent Frobenius series solutions on (0,).2x2 y x(1 + 2x)y + 2(4x 1)y = 0. 1
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Chapter 11: Problem 16 Differential Equations and Linear Algebra 4
For Problems 1017, show that the indicial equation of the given differential equation has distinct roots that do not differ by an integer and find two linearly independent Frobenius series solutions on (0,).x2 y + x(1 x)y (5 + x)y = 0. 1
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Chapter 11: Problem 17 Differential Equations and Linear Algebra 4
For Problems 1017, show that the indicial equation of the given differential equation has distinct roots that do not differ by an integer and find two linearly independent Frobenius series solutions on (0,).3x2 y + x(7 + 3x)y + (1 + 6x)y = 0. 1
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Chapter 11: Problem 18 Differential Equations and Linear Algebra 4
Consider the differential equation x2 y + x y + (1 x)y = 0, x > 0. (11.4.15) (a) Find the indicial equation, and show that the roots are r = i. (b) Determine the first three terms in a complexvalued Frobenius series solution to Equation (11.4.15). (c) Use the solution in (b) to determine two linearly independent real-valued solutions to Equation (11.4.15). 1
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Chapter 11: Problem 19 Differential Equations and Linear Algebra 4
Determine the first five nonzero terms in each of two linearly independent Frobenius series solutions to 3x2 y + x(1 + 3x2)y 2x y = 0, x > 0. 2
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Chapter 11: Problem 20 Differential Equations and Linear Algebra 4
Consider the differential equation 4x2 y 4x2 y + (1 + 2x)y = 0. (a) Show that the indicial equation has only one root, and find the corresponding Frobenius series solution. (b) Use the reduction of order technique to find a second linearly independent solution on (0,). [Hint: To evaluate ex x dx, expand ex in a Maclaurin series.] 2
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Chapter 11: Problem 21 Differential Equations and Linear Algebra 4
Find two linearly independent solutions to x2 y + x(3 2x)y + (1 2x)y = 0 on (0,).
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