Find the general solutions of the differential equations in Problems 1 through 20.
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1
First-Order Differential Equations
1.1
First-Order Differential Equations
1.2
Integrals as General and Particular Solutions
1.3
Slope Fields and Solution Curves
1.4
Separable Equations and Applications
1.5
Linear First-Order Equations
1.6
Substitution Methods and Exact Equations
2.1
Population Models
2.2
Equilibrium Solutions and Stability
2.3
AccelerationVelocity Models
2.4
Numerical Approximation: Eulers Method
2.5
A Closer Look at the Euler Method
2.6
The RungeKutta Method
3.1
Introduction: Second-Order Linear Equations
3.2
General Solutions of Linear Equations
3.3
Homogeneous Equations with Constant Coefficients
3.4
Mechanical Vibrations
3.5
Nonhomogeneous Equations and Undetermined Coefficients
3.6
Forced Oscillations and Resonance
3.7
Electrical Circuits
3.8
Mathematical Models and Numerical Methods
4.1
First-Order Systems and Applications
4.2
The Method of Elimination
4.3
Numerical Methods for Systems
5.1
Matrices and Linear Systems
5.2
The Eigenvalue Method for Homogeneous Systems
5.3
A Gallery of Solution Curves of Linear Systems
5.4
Second-Order Systems and Mechanical Applications
5.5
Multiple Eigenvalue Solutions
5.6
Matrix Exponentials and Linear Systems
5.7
Nonhomogeneous Linear Systems
6.1
Stability and the Phase Plane
6.2
Linear and Almost Linear Systems
6.3
Ecological Models: Predators and Competitors
6.4
Nonlinear Mechanical Systems
7.1
Laplace Transforms and Inverse Transforms
7.2
Transformation of Initial Value Problems
7.3
Translation and Partial Fractions
7.4
Derivatives, Integrals, and Products of Transforms
7.5
Periodic and Piecewise Continuous Input Functions
7.6
Laplace Transform Methods
8.1
Introduction and Review of Power Series
8.2
Series Solutions Near Ordinary Points
8.3
Regular Singular Points
8.4
Method of Frobenius: The Exceptional Cases
8.5
Bessels Equation
8.6
Applications of Bessel Functions
9.1
Periodic Functions and Trigonometric Series
9.2
General Fourier Series and Convergence
9.3
Fourier Sine and Cosine Series
9.4
Applications of Fourier Series
9.5
Heat Conduction and Separation of Variables
9.6
Vibrating Strings and the One-Dimensional Wave Equation
9.7
Fourier Series Methods and Partial Differential Equations
10.1
SturmLiouville Problems and Eigenfunction Expansions
10.2
Applications of Eigenfunction Series
10.3
Steady Periodic Solutions and Natural Frequencies
10.4
Cylindrical Coordinate Problems
Textbook Solutions for Differential Equations and Boundary Value Problems: Computing and Modeling
Chapter 3.3 Problem 52
Question
Make the substitution v D ln x of to find general solutions (for x>0) of the Euler equations in 52 through 58.
Solution
The first step in solving 3.3 problem number 52 trying to solve the problem we have to refer to the textbook question: Make the substitution v D ln x of to find general solutions (for x>0) of the Euler equations in 52 through 58.
From the textbook chapter Homogeneous Equations with Constant Coefficients you will find a few key concepts needed to solve this.
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full solution
Title
Differential Equations and Boundary Value Problems: Computing and Modeling 5
Author
C. Henry Edwards, David E. Penney, David T. Calvis
ISBN
9780321796981
Solved: Make the substitution v D ln x of to find general
Chapter 3.3 textbook questions
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Chapter 3: Problem 1 Differential Equations and Boundary Value Problems: Computing and Modeling 5
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Chapter 3: Problem 2 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find the general solutions of the differential equations in Problems 1 through 20.
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Chapter 3: Problem 3 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find the general solutions of the differential equations in Problems 1 through 20.
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Chapter 3: Problem 4 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find the general solutions of the differential equations in Problems 1 through 20.
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Chapter 3: Problem 5 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find the general solutions of the differential equations in Problems 1 through 20.
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Chapter 3: Problem 6 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find the general solutions of the differential equations in Problems 1 through 20.
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Chapter 3: Problem 7 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find the general solutions of the differential equations in Problems 1 through 20.
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Chapter 3: Problem 8 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find the general solutions of the differential equations in Problems 1 through 20.
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Chapter 3: Problem 9 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find the general solutions of the differential equations in Problems 1 through 20.
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Chapter 3: Problem 10 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find the general solutions of the differential equations in Problems 1 through 20.
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Chapter 3: Problem 11 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find the general solutions of the differential equations in Problems 1 through 20.
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Chapter 3: Problem 12 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find the general solutions of the differential equations in Problems 1 through 20.
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Chapter 3: Problem 13 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find the general solutions of the differential equations in Problems 1 through 20.
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Chapter 3: Problem 14 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find the general solutions of the differential equations in Problems 1 through 20.
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Chapter 3: Problem 15 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find the general solutions of the differential equations in Problems 1 through 20.
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Chapter 3: Problem 16 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find the general solutions of the differential equations in Problems 1 through 20.
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Chapter 3: Problem 17 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find the general solutions of the differential equations in Problems 1 through 20.
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Chapter 3: Problem 18 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find the general solutions of the differential equations in Problems 1 through 20.
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Chapter 3: Problem 19 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find the general solutions of the differential equations in Problems 1 through 20.
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Chapter 3: Problem 20 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find the general solutions of the differential equations in Problems 1 through 20.
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Chapter 3: Problem 21 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Solve the initial value problems given in Problems 21 through 26.
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Chapter 3: Problem 22 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Solve the initial value problems given in Problems 21 through 26.
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Chapter 3: Problem 23 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Solve the initial value problems given in Problems 21 through 26.
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Chapter 3: Problem 24 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Solve the initial value problems given in Problems 21 through 26.
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Chapter 3: Problem 25 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Solve the initial value problems given in Problems 21 through 26.
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Chapter 3: Problem 26 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Solve the initial value problems given in Problems 21 through 26.
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Chapter 3: Problem 27 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find general solutions of the equations in Problems 27 through 32. First find a small integral root of the characteristic equation by inspection; then factor by division.
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Chapter 3: Problem 28 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find general solutions of the equations in Problems 27 through 32. First find a small integral root of the characteristic equation by inspection; then factor by division.
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Chapter 3: Problem 29 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find general solutions of the equations in Problems 27 through 32. First find a small integral root of the characteristic equation by inspection; then factor by division.
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Chapter 3: Problem 30 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find general solutions of the equations in Problems 27 through 32. First find a small integral root of the characteristic equation by inspection; then factor by division.
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Chapter 3: Problem 31 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find general solutions of the equations in Problems 27 through 32. First find a small integral root of the characteristic equation by inspection; then factor by division.
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Chapter 3: Problem 32 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find general solutions of the equations in Problems 27 through 32. First find a small integral root of the characteristic equation by inspection; then factor by division.
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Chapter 3: Problem 33 Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 33 through 36, one solution of the differential equation is given. Find the general solution.
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Chapter 3: Problem 34 Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 33 through 36, one solution of the differential equation is given. Find the general solution.
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Chapter 3: Problem 35 Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 33 through 36, one solution of the differential equation is given. Find the general solution.
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Chapter 3: Problem 36 Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 33 through 36, one solution of the differential equation is given. Find the general solution.
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Chapter 3: Problem 37 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find a function y.x/ such that y.4/.x/ D y.3/.x/ for all x and y.0/ D 18, y0 .0/ D 12, y00.0/ D 13, and y.3/.0/ D 7.
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Chapter 3: Problem 38 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Solve the initial value problem y.3/ 5y00 C 100y0 500y D 0I y.0/ D 0; y0 .0/ D 10; y00.0/ D 250 given that y1.x/ D e5x is one particular solution of the differential equation.
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Chapter 3: Problem 39 Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 39 through 42, find a linear homogeneous constant-coefficient equation with the given general solution.
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Chapter 3: Problem 40 Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 39 through 42, find a linear homogeneous constant-coefficient equation with the given general solution.
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Chapter 3: Problem 41 Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 39 through 42, find a linear homogeneous constant-coefficient equation with the given general solution.
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Chapter 3: Problem 42 Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 39 through 42, find a linear homogeneous constant-coefficient equation with the given general solution.
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Chapter 3: Problem 43 Differential Equations and Boundary Value Problems: Computing and Modeling 5
(a) Use Eulers formula to show that every complex number can be written in the form rei, where r = 0 and <5 . (b) Express the numbers 4, 2, 3i, 1 C i, and 1 C i p3 in the form rei. (c) The two square roots of reiare prei=2. Find the square roots of the numbers 2 2ip3 and 2 C 2ip3. 44. U
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Chapter 3: Problem 44 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Use the quadratic formula to solve the following equations. Note in each case that the roots are not complex conjugates. (a) x2 C ix C 2 D 0 (b) x2 2ix C 3 D 0
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Chapter 3: Problem 45 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find a general solution of y00 2iy0 C 3y D 0.
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Chapter 3: Problem 46 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find a general solution of y00 iy0 C 6y D 0. 47. Find a general solution of y00 D 2 C 2ip3 y.
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Chapter 3: Problem 47 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find a general solution of y00 D 2 C 2ip3 y.
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Chapter 3: Problem 48 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Solve the initial value problem y.3/ D yI y.0/ D 1; y0 .0/ D y00.0/ D 0: (Suggestion: Impose the given initial conditions on the general solution y.x/ D Aex C Bex C Cex
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Chapter 3: Problem 49 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Solve the initial value problem y.4/ D y.3/ C y00 C y0 C 2yI y.0/ D y0 .0/ D y00.0/ D 0; 2y.3/.0/ D 30
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Chapter 3: Problem 50 Differential Equations and Boundary Value Problems: Computing and Modeling 5
The differential equation y00 C .sgn x/y D 0 (25) has the discontinuous coefficient function sgn x D ( C1 if x>0, 1 if x<0. Show that Eq. (25) nevertheless has two linearly independent solutions y1.x/ and y2.x/ defined for all x such that Each satisfies Eq. (25) at each point x 0; Each has a continuous derivative at x D 0; y1.0/ D y0 2.0/ D 1 and y2.0/ D y0 1.0/ D 0. (Suggestion: Each yi.x/ will be defined by one formula for x<0 and by another for x = 0.) The graphs of these two solutions are shown in Fig. 3.3.2. 2
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Chapter 3: Problem 51 Differential Equations and Boundary Value Problems: Computing and Modeling 5
The differential equation y00 C .sgn x/y D 0 (25) has the discontinuous coefficient function sgn x D ( C1 if x>0, 1 if x<0. Show that Eq. (25) nevertheless has two linearly independent solutions y1.x/ and y2.x/ defined for all x such that Each satisfies Eq. (25) at each point x 0; Each has a continuous derivative at x D 0; y1.0/ D y0 2.0/ D 1 and y2.0/ D y0 1.0/ D 0. (Suggestion: Each yi.x/ will be defined by one formula for x<0 and by another for x = 0.) The graphs of these two solutions are shown in Fig. 3.3.2. 2
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Chapter 3: Problem 52 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Make the substitution v D ln x of Problem 51 to find general solutions (for x>0) of the Euler equations in Problems 52 through 58.
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Chapter 3: Problem 53 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Make the substitution v D ln x of Problem 51 to find general solutions (for x>0) of the Euler equations in Problems 52 through 58.
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Chapter 3: Problem 54 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Make the substitution v D ln x of Problem 51 to find general solutions (for x>0) of the Euler equations in Problems 52 through 58.
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Chapter 3: Problem 55 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Make the substitution v D ln x of Problem 51 to find general solutions (for x>0) of the Euler equations in Problems 52 through 58.
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Chapter 3: Problem 56 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Make the substitution v D ln x of Problem 51 to find general solutions (for x>0) of the Euler equations in Problems 52 through 58.
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Chapter 3: Problem 57 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Make the substitution v D ln x of Problem 51 to find general solutions (for x>0) of the Euler equations in Problems 52 through 58.
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Chapter 3: Problem 58 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Make the substitution v D ln x of Problem 51 to find general solutions (for x>0) of the Euler equations in Problems 52 through 58.
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