In Problems 1 through 16, transform the given differential equation or system into an equivalent system of first-order differential equations
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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 4.1 Problem 32
Question
Three 100-gal fermentation vats are connected as indicated in Fig. 4.1.13, and the mixture in each tank is kept uniform by stirring. Denote by xi.t / the amount (in pounds) of alcohol in tank Ti at time t (i D 1, 2, 3). Suppose that the mixture circulates between the tanks at the rate of 10 gal=min. Derive the equations 10x0 1 D x1 C x3 10x0 2 D x1 x2 10x0 3 D x2 x3. T
Solution
The first step in solving 4.1 problem number 32 trying to solve the problem we have to refer to the textbook question: Three 100-gal fermentation vats are connected as indicated in Fig. 4.1.13, and the mixture in each tank is kept uniform by stirring. Denote by xi.t / the amount (in pounds) of alcohol in tank Ti at time t (i D 1, 2, 3). Suppose that the mixture circulates between the tanks at the rate of 10 gal=min. Derive the equations 10x0 1 D x1 C x3 10x0 2 D x1 x2 10x0 3 D x2 x3. T
From the textbook chapter First-Order Systems and Applications you will find a few key concepts needed to solve this.
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Title
Differential Equations and Boundary Value Problems: Computing and Modeling 5
Author
C. Henry Edwards, David E. Penney, David T. Calvis
ISBN
9780321796981
Three 100-gal fermentation vats are connected as indicated
Chapter 4.1 textbook questions
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Chapter 4: Problem 1 Differential Equations and Boundary Value Problems: Computing and Modeling 5
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Chapter 4: Problem 2 Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 1 through 16, transform the given differential equation or system into an equivalent system of first-order differential equations
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Chapter 4: Problem 3 Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 1 through 16, transform the given differential equation or system into an equivalent system of first-order differential equations
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Chapter 4: Problem 4 Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 1 through 16, transform the given differential equation or system into an equivalent system of first-order differential equations
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Chapter 4: Problem 5 Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 1 through 16, transform the given differential equation or system into an equivalent system of first-order differential equations
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Chapter 4: Problem 6 Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 1 through 16, transform the given differential equation or system into an equivalent system of first-order differential equations
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Chapter 4: Problem 7 Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 1 through 16, transform the given differential equation or system into an equivalent system of first-order differential equations
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Chapter 4: Problem 8 Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 1 through 16, transform the given differential equation or system into an equivalent system of first-order differential equations
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Chapter 4: Problem 9 Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 1 through 16, transform the given differential equation or system into an equivalent system of first-order differential equations
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Chapter 4: Problem 10 Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 1 through 16, transform the given differential equation or system into an equivalent system of first-order differential equations
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Chapter 4: Problem 11 Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 1 through 16, transform the given differential equation or system into an equivalent system of first-order differential equations
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Chapter 4: Problem 12 Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 1 through 16, transform the given differential equation or system into an equivalent system of first-order differential equations
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Chapter 4: Problem 13 Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 1 through 16, transform the given differential equation or system into an equivalent system of first-order differential equations
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Chapter 4: Problem 14 Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 1 through 16, transform the given differential equation or system into an equivalent system of first-order differential equations
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Chapter 4: Problem 15 Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 1 through 16, transform the given differential equation or system into an equivalent system of first-order differential equations
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Chapter 4: Problem 16 Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 1 through 16, transform the given differential equation or system into an equivalent system of first-order differential equations
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Chapter 4: Problem 17 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Use the method of Examples 6, 7, and 8 to find general solutions of the systems in Problems 17 through 26. If initial conditions are given, find the corresponding particular solution. For each problem, use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system.
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Chapter 4: Problem 18 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Use the method of Examples 6, 7, and 8 to find general solutions of the systems in Problems 17 through 26. If initial conditions are given, find the corresponding particular solution. For each problem, use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system.
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Chapter 4: Problem 19 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Use the method of Examples 6, 7, and 8 to find general solutions of the systems in Problems 17 through 26. If initial conditions are given, find the corresponding particular solution. For each problem, use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system.
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Chapter 4: Problem 20 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Use the method of Examples 6, 7, and 8 to find general solutions of the systems in Problems 17 through 26. If initial conditions are given, find the corresponding particular solution. For each problem, use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system.
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Chapter 4: Problem 21 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Use the method of Examples 6, 7, and 8 to find general solutions of the systems in Problems 17 through 26. If initial conditions are given, find the corresponding particular solution. For each problem, use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system.
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Chapter 4: Problem 22 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Use the method of Examples 6, 7, and 8 to find general solutions of the systems in Problems 17 through 26. If initial conditions are given, find the corresponding particular solution. For each problem, use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system.
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Chapter 4: Problem 23 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Use the method of Examples 6, 7, and 8 to find general solutions of the systems in Problems 17 through 26. If initial conditions are given, find the corresponding particular solution. For each problem, use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system.
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Chapter 4: Problem 24 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Use the method of Examples 6, 7, and 8 to find general solutions of the systems in Problems 17 through 26. If initial conditions are given, find the corresponding particular solution. For each problem, use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system.
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Chapter 4: Problem 25 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Use the method of Examples 6, 7, and 8 to find general solutions of the systems in Problems 17 through 26. If initial conditions are given, find the corresponding particular solution. For each problem, use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system.
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Chapter 4: Problem 26 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Use the method of Examples 6, 7, and 8 to find general solutions of the systems in Problems 17 through 26. If initial conditions are given, find the corresponding particular solution. For each problem, use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system.
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Chapter 4: Problem 27 Differential Equations and Boundary Value Problems: Computing and Modeling 5
(a) Calculate x.t /2 C y.t /2 to show that the trajectories of the system x0 D y, y0 D x of Problem 17 are circles. (b) Calculate x.t /2 y.t /2 to show that the trajectories of the system x0 D y, y0 D x of Problem 18 are hyperbolas. 28.
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Chapter 4: Problem 28 Differential Equations and Boundary Value Problems: Computing and Modeling 5
(a) Beginning with the general solution of the system x0 D 2y, y0 D 2x of Problem 19, calculate x2 C y2 to show that the trajectories are circles. (b) Show similarly that the trajectories of the system x0 D 1 2 y, y0 D 8x of Problem 21 are ellipses with equations of the form 16x2 C y2 D C2.
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Chapter 4: Problem 29 Differential Equations and Boundary Value Problems: Computing and Modeling 5
First solve Eqs. (20) and (21) for et and e2t in terms of x.t /, y.t /, and the constants A and B. Then substitute the results in .e2t/.et/2 D 1 to show that the trajectories of the system x0 D y, y0 D 2x C y in Example 7 satisfy an equation of the form 4x3 3xy2 C y3 D C (constant): Then show that C D 0 yields the straight lines y D x and y D 2x that are visible in Fig. 4.1.8. 30
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Chapter 4: Problem 30 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Derive the equations m1x00 1 D .k1 C k2/x1 C k2x2, m2x00 2 D k2x1 .k2 C k3/x2
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Chapter 4: Problem 31 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Two particles each of mass m are attached to a string under (constant) tension T , as indicated in Fig. 4.1.12. Assume that the particles oscillate vertically (that is, parallel to the y-axis) with amplitudes so small that the sines of the angles shown are accurately approximated by their tangents. Show that the displacements y1 and y2 satisfy the equations ky00 1 D 2y1 C y2; ky00 2 D y1 2y2
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Chapter 4: Problem 32 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Three 100-gal fermentation vats are connected as indicated in Fig. 4.1.13, and the mixture in each tank is kept uniform by stirring. Denote by xi.t / the amount (in pounds) of alcohol in tank Ti at time t (i D 1, 2, 3). Suppose that the mixture circulates between the tanks at the rate of 10 gal=min. Derive the equations 10x0 1 D x1 C x3 10x0 2 D x1 x2 10x0 3 D x2 x3. T
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Chapter 4: Problem 33 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Set up a system of first-order differential equations for the indicated currents I1 and I2 in the electrical circuit of Fig. 4.1.14, which shows an inductor, two resistors, and a generator which supplies an alternating voltage drop of E.t / D 100 sin 60t V in the direction of the current I1.
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Chapter 4: Problem 34 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Repeat Problem 33, except with the generator replaced with a battery supplying an emf of 100 V and with the inductor replaced with a 1-millifarad (mF) capacitor
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Chapter 4: Problem 35 Differential Equations and Boundary Value Problems: Computing and Modeling 5
A particle of mass m moves in the plane with coordinates .x.t /; y.t // under the influence of a force that is directed toward the origin and has magnitude k=.x2 C y2/an inverse-square central force field. Show that mx00 D kx r3 and my00 D ky r3 ; where r D px2 C y2.
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Chapter 4: Problem 36 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Suppose that a projectile of mass m moves in a vertical plane in the atmosphere near the surface of the earth under the influence of two forces: a downward gravitationa
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Chapter 4: Problem 37 Differential Equations and Boundary Value Problems: Computing and Modeling 5
Suppose that a particle with mass m and electrical charge q moves in the xy-plane under the influence of the magnetic field B D Bk (thus a uniform field parallel to the -axis), so the force on the particle is F D qv B if its velocity is v. Show that the equations of motion of the particle are mx00 D CqBy0 ; my00 D qBx
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