In each of Problems 1 through 12 find the general solution of the given system of equations.x = 2 1 3 2 x + et t
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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 7.9 Problem 4
Question
In each of 1 through 12 find the general solution of the given system of equations.x =1 14 2x +e2t2et
Solution
The first step in solving 7.9 problem number 4 trying to solve the problem we have to refer to the textbook question: In each of 1 through 12 find the general solution of the given system of equations.x =1 14 2x +e2t2et
From the textbook chapter Nonhomogeneous Linear Systems you will find a few key concepts needed to solve this.
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full solution
full solution
Title
Elementary Differential Equations and Boundary Value Problems 9
Author
Boyce, Richard C. DiPrima
ISBN
9780470383346
In each of 1 through 12 find the general solution of the given system of equations.x =1
Chapter 7.9 textbook questions
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Chapter 7: Problem 1 Elementary Differential Equations and Boundary Value Problems 9
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Chapter 7: Problem 2 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 12 find the general solution of the given system of equations.x = 1 3 3 1 x + et 3 et
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Chapter 7: Problem 3 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 12 find the general solution of the given system of equations.x = 2 5 1 2 x + cost sin t
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Chapter 7: Problem 4 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 12 find the general solution of the given system of equations.x = 1 1 4 2 x + e2t 2et
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Chapter 7: Problem 5 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 12 find the general solution of the given system of equations.x = 4 2 8 4 x + t 3 t 2 , t > 0
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Chapter 7: Problem 6 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 12 find the general solution of the given system of equations.x = 4 2 2 1 x + t 1 2t 1 + 4 , t > 0
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Chapter 7: Problem 7 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 12 find the general solution of the given system of equations.x = 1 1 4 1 x + 2 1 e
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Chapter 7: Problem 8 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 12 find the general solution of the given system of equations.x = 2 1 3 2 x + 1 1 et
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Chapter 7: Problem 9 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 12 find the general solution of the given system of equations. = 5 4 3 4 3 4 5 4 x + 2t et
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Chapter 7: Problem 10 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 12 find the general solution of the given system of equations.x = 3 2 2 2 x + 1 1 et
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Chapter 7: Problem 11 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 12 find the general solution of the given system of equations.. x = 2 5 1 2 x + 0 cost , 0 < t <
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Chapter 7: Problem 12 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 12 find the general solution of the given system of equations.x = 2 5 1 2 x + csc t sec t , 2 < t <
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Chapter 7: Problem 13 Elementary Differential Equations and Boundary Value Problems 9
The electric circuit shown in Figure 7.9.1 is described by the system of differential equations dx dt = 1 2 1 8 2 1 2 x + 1 2 0 I(t), (i) where x1 is the current through the inductor, x2 is the voltage drop across the capacitor, and I(t) is the current supplied by the external source. I(t) R = 4 ohms R = 4 ohms L = 8 henrys C = farad 1 2 FIGURE 7.9.1 The circuit in Problem 13. (a) Determine a fundamental matrix (t) for the homogeneous system corresponding to Eq. (i). Refer to Problem 25 of Section 7.6. (b) If I(t) = et/2, determine the solution of the system (i) that also satisfies the initial conditions x(0) = 0.
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Chapter 7: Problem 14 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 14 and 15 verify that the given vector is the general solution of the corresponding homogeneous system, and then solve the nonhomogeneous system. Assume that t > 0.
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Chapter 7: Problem 15 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 14 and 15 verify that the given vector is the general solution of the corresponding homogeneous system, and then solve the nonhomogeneous system. Assume that t > 0.
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Chapter 7: Problem 16 Elementary Differential Equations and Boundary Value Problems 9
Let x = (t) be the general solution of x = P(t)x + g(t), and let x = v(t) be some particular solution of the same system. By considering the difference (t) v(t), show that (t) = u(t) + v(t), where u(t) is the general solution of the homogeneous system x = P(t)x
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Chapter 7: Problem 17 Elementary Differential Equations and Boundary Value Problems 9
Consider the initial value problem x = Ax + g(t), x(0) = x0 . (a) By referring to Problem 15(c) in Section 7.7, show that x = (t)x0 + t 0 (t s)g(s) ds. (b) Show also that x = exp(At)x0 + t 0 exp[A(t s)]g(s) ds. Compare these results with those of Problem 27 in Section 3.6.
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