In Problems 112 use the Laplace transform to solve the given system of differential equations.
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Table of Contents
1
INTRODUCTION TO DIFFERENTIAL EQUATIONS
1.1
Definitions and Terminology
1.2
Initial-Value Problems
1.3
Differential Equations as Mathematical Models
2
FIRST-ORDER DIFFERENTIAL EQUATIONS
2.1
Solution Curves Without a Solution
2.2
Separable Variables
2.3
Linear Equations
2.4
Exact Equations
2.5
Solutions by Substitutions
2.6
A Numerical Method
3
MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS
3.1
Linear Models
3.2
Nonlinear Models
3.3
Modeling with Systems of First-Order DEs
4
HIGHER-ORDER DIFFERENTIAL EQUATIONS
4.1
Preliminary TheoryLinear Equations
4.2
Reduction of Order
4.3
Homogeneous Linear Equations with Constant Coefficients
4.4
Undetermined CoefficientsSuperposition Approach
4.5
Undetermined CoefficientsAnnihilator Approach
4.6
Variation of Parameters
4.7
Cauchy-Euler Equation
4.8
Solving Systems of Linear DEs by Elimination
4.9
Nonlinear Differential Equations
5
MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS
5.1
Linear Models: Initial-Value Problems
5.2
Linear Models: Boundary-Value Problems
5.3
Nonlinear Models
6
SERIES SOLUTIONS OF LINEAR EQUATIONS
6.1
Solutions About Ordinary Points
6.2
Solutions About Singular Points
6.3
Special Functions
7
THE LAPLACE TRANSFORM
7.1
Definition of the Laplace Transform
7.2
Inverse Transforms and Transforms of Derivatives
7.3
Operational Properties I
7.4
Operational Properties II
7.5
The Dirac Delta Function
7.6
Systems of Linear Differential Equations
8
SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS
8.1
Preliminary TheoryLinear Systems
8.2
Homogeneous Linear Systems
8.3
Nonhomogeneous Linear Systems
8.4
Matrix Exponential
9
NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS
9.1
Euler Methods and Error Analysis
9.2
Runge-Kutta Methods
9.3
Multistep Methods
9.4
Higher-Order Equations and Systems
9.5
Second-Order Boundary-Value Problems
10
PLANE AUTONOMOUS SYSTEMS
10.1
Autonomous Systems
10.2
Stability of Linear Systems
10.3
Linearization and Local Stability
10.4
Autonomous Systems as Mathematical Models
11
ORTHOGONAL FUNCTIONS AND FOURIER SERIES
11.1
Orthogonal Functions
11.2
Fourier Series
11.3
Fourier Cosine and Sine Series
11.4
Sturm-Liouville Problem
11.5
Bessel and Legendre Series
12
BOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES
12.1
Separable Partial Differential Equations
12.2
Classical PDEs and Boundary-Value Problems
12.3
Heat Equation
12.4
Wave Equation
12.5
Laplaces Equation
12.6
Nonhomogeneous Boundary-Value Problems
12.7
Orthogonal Series Expansions
12.8
Higher-Dimensional Problems
13
BOUNDARY-VALUE PROBLEMS IN OTHER COORDINATE SYSTEMS
13.1
Polar Coordinates
13.2
Polar and Cylindrical Coordinates
13.3
Spherical Coordinates
14
INTEGRAL TRANSFORMS
14.1
Error Function
14.2
Laplace Transform
14.3
Fourier Integral
14.4
Fourier Transforms
15
NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
15.1
Laplaces Equation
15.2
Heat Equation
15.3
Wave Equation
Textbook Solutions for Differential Equations with Boundary-Value Problems
Chapter 7.6 Problem 21
Question
(a) Use the Laplace transform and the information given in Example 3 to obtain the solution (8) of the system given in (7). (b) Use a graphing utility to graph u1(t) and u2(t) in the tu-plane. Which mass has extreme displacements of greater magnitude? Use the graphs to estimate the first time that each mass passes through its equilibrium position. Discuss whether the motion of the pendulums is periodic. (c) Graph u1(t) and u2(t) in the u1u2-plane as parametric equations. The curve defined by these parametric equations is called a Lissajous curve. (d) The positions of the masses at t 0 are given in Figure 7.6.5(a). Note that we have used 1 radian 57.3. Use a calculator or a table application in a CAS to construct a table of values of the angles u1 and u2 for t 1, 2, . . . , 10 s. Then plot the positions of the two masses at these times. (e) Use a CAS to find the first time that u1(t) u2(t) and compute the corresponding angular value. Plot the positions of the two masses at these times. (f) Utilize the CAS to draw appropriate lines to simulate the pendulum rods, as in Figure 7.6.5. Use the animation capability of your CAS to make a movie of the motion of the double pendulum from t 0 to t 10 using a time increment of 0.1. [Hint: Express the coordinates (x1(t), y1(t)) and (x2(t), y2(t)) of the masses m1 and m2, respectively, in terms of u1(t) and u2(t).]
Solution
The first step in solving 7.6 problem number 21 trying to solve the problem we have to refer to the textbook question: (a) Use the Laplace transform and the information given in Example 3 to obtain the solution (8) of the system given in (7). (b) Use a graphing utility to graph u1(t) and u2(t) in the tu-plane. Which mass has extreme displacements of greater magnitude? Use the graphs to estimate the first time that each mass passes through its equilibrium position. Discuss whether the motion of the pendulums is periodic. (c) Graph u1(t) and u2(t) in the u1u2-plane as parametric equations. The curve defined by these parametric equations is called a Lissajous curve. (d) The positions of the masses at t 0 are given in Figure 7.6.5(a). Note that we have used 1 radian 57.3. Use a calculator or a table application in a CAS to construct a table of values of the angles u1 and u2 for t 1, 2, . . . , 10 s. Then plot the positions of the two masses at these times. (e) Use a CAS to find the first time that u1(t) u2(t) and compute the corresponding angular value. Plot the positions of the two masses at these times. (f) Utilize the CAS to draw appropriate lines to simulate the pendulum rods, as in Figure 7.6.5. Use the animation capability of your CAS to make a movie of the motion of the double pendulum from t 0 to t 10 using a time increment of 0.1. [Hint: Express the coordinates (x1(t), y1(t)) and (x2(t), y2(t)) of the masses m1 and m2, respectively, in terms of u1(t) and u2(t).]
From the textbook chapter Systems of Linear Differential Equations you will find a few key concepts needed to solve this.
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full solution
Title
Differential Equations with Boundary-Value Problems 7
Author
Dennis G. Zill, Michael R. Cullen
ISBN
9780495108368
Solved: (a) Use the Laplace transform and the information given in Example 3 to obtain
Chapter 7.6 textbook questions
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Chapter 7: Problem 1 Differential Equations with Boundary-Value Problems 7
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Chapter 7: Problem 2 Differential Equations with Boundary-Value Problems 7
In Problems 112 use the Laplace transform to solve the given system of differential equations.
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Chapter 7: Problem 3 Differential Equations with Boundary-Value Problems 7
In Problems 112 use the Laplace transform to solve the given system of differential equations.
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Chapter 7: Problem 4 Differential Equations with Boundary-Value Problems 7
In Problems 112 use the Laplace transform to solve the given system of differential equations.
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Chapter 7: Problem 5 Differential Equations with Boundary-Value Problems 7
In Problems 112 use the Laplace transform to solve the given system of differential equations.
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Chapter 7: Problem 6 Differential Equations with Boundary-Value Problems 7
In Problems 112 use the Laplace transform to solve the given system of differential equations.
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Chapter 7: Problem 7 Differential Equations with Boundary-Value Problems 7
In Problems 112 use the Laplace transform to solve the given system of differential equations.
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Chapter 7: Problem 8 Differential Equations with Boundary-Value Problems 7
In Problems 112 use the Laplace transform to solve the given system of differential equations.
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Chapter 7: Problem 9 Differential Equations with Boundary-Value Problems 7
In Problems 112 use the Laplace transform to solve the given system of differential equations.
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Chapter 7: Problem 10 Differential Equations with Boundary-Value Problems 7
In Problems 112 use the Laplace transform to solve the given system of differential equations.
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Chapter 7: Problem 11 Differential Equations with Boundary-Value Problems 7
In Problems 112 use the Laplace transform to solve the given system of differential equations.
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Chapter 7: Problem 12 Differential Equations with Boundary-Value Problems 7
In Problems 112 use the Laplace transform to solve the given system of differential equations.
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Chapter 7: Problem 13 Differential Equations with Boundary-Value Problems 7
Solve system (1) when k1 3, k2 2, m1 1, m2 1 and x1(0) 0, , x2(0) 1, .
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Chapter 7: Problem 14 Differential Equations with Boundary-Value Problems 7
Derive the system of differential equations describing the straight-line vertical motion of the coupled springs shown in Figure 7.6.6. Use the Laplace transform to solve the system when k1 1, k2 1, k3 1, m1 1, m2 1 and x1(0) 0, , x2(0) 0, .
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Chapter 7: Problem 15 Differential Equations with Boundary-Value Problems 7
(a) Show that the system of differential equations for the currents i2(t) and i3(t) in the electrical network shown in Figure 7.6.7 is (b) Solve the system in part (a) if R 5 ", L1 0.01 h, L2 0.0125 h, E 100 V, i2(0) 0, and i3(0) 0. (c) Determine the current i1(t).
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Chapter 7: Problem 16 Differential Equations with Boundary-Value Problems 7
(a) In Problem 12 in Exercises 3.3 you were asked to show that the currents i2(t) and i3(t) in the electrical network shown in Figure 7.6.8 satisfy Solve the system if R1 10 ", R2 5 ", L 1 h, C 0.2 f, i2(0) 0, and i3(0) 0. (b) Determine the current i1(t).
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Chapter 7: Problem 17 Differential Equations with Boundary-Value Problems 7
Solve the system given in (17) of Section 3.3 when R1 6 ", R2 5", L1 1 h, L2 1 h, E(t) 50 sin t V, i2(0) 0, and i3(0) 0.
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Chapter 7: Problem 18 Differential Equations with Boundary-Value Problems 7
Solve (5) when E 60 V, , R 50 ", C 104 f, i1(0) 0, and i2(0) 0.
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Chapter 7: Problem 19 Differential Equations with Boundary-Value Problems 7
Solve (5) when E 60 V, L 2 h, R 50 ", C 104 f, i1(0) 0, and i2(0) 0.
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Chapter 7: Problem 20 Differential Equations with Boundary-Value Problems 7
(a) Show that the system of differential equations for the charge on the capacitor q(t) and the current i3(t) in the electrical network shown in Figure 7.6.9 is (b) Find the charge on the capacitor when L 1 h, R1 1 ", R2 1 ", C 1 f, i3(0) 0, and q(0) 0. E(t) 0, 50et ,
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Chapter 7: Problem 21 Differential Equations with Boundary-Value Problems 7
(a) Use the Laplace transform and the information given in Example 3 to obtain the solution (8) of the system given in (7). (b) Use a graphing utility to graph u1(t) and u2(t) in the tu-plane. Which mass has extreme displacements of greater magnitude? Use the graphs to estimate the first time that each mass passes through its equilibrium position. Discuss whether the motion of the pendulums is periodic. (c) Graph u1(t) and u2(t) in the u1u2-plane as parametric equations. The curve defined by these parametric equations is called a Lissajous curve. (d) The positions of the masses at t 0 are given in Figure 7.6.5(a). Note that we have used 1 radian 57.3. Use a calculator or a table application in a CAS to construct a table of values of the angles u1 and u2 for t 1, 2, . . . , 10 s. Then plot the positions of the two masses at these times. (e) Use a CAS to find the first time that u1(t) u2(t) and compute the corresponding angular value. Plot the positions of the two masses at these times. (f) Utilize the CAS to draw appropriate lines to simulate the pendulum rods, as in Figure 7.6.5. Use the animation capability of your CAS to make a movie of the motion of the double pendulum from t 0 to t 10 using a time increment of 0.1. [Hint: Express the coordinates (x1(t), y1(t)) and (x2(t), y2(t)) of the masses m1 and m2, respectively, in terms of u1(t) and u2(t).]
Read more