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3. Is the differential equation derived in Problems 1 and 2 linear or nonlinear? Explain why.

3. Is the differential equation derived in Problems 1 and 2 linear or nonlinear? Explain why.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.7: Applications
Problem 18EQ
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10:14 PM B Select text O: Select all Copy 1. Assuming that both springs have spring constant k and that there is a damping force proportional to velocity x' with damping constant y, write down the differential equation that describes the motion of the mass. Note that the expression for the force function depends on which of the three different parts of its domain x lies within. tions x(0) = 2, x'(0) = 0. Find the solution x(t) of the initial value problem for 0<1< and si<5, where t and t; are the times of the first and second events. Describe the physical situation corresponding to each of these two events. 6. Consider the damped problem using the parameter values L = 1, m = 1, y = 0.1, and k = 1. 2. The Heaviside, or unit step function, is defined by (a) Use a computer to draw a direction field of the corre- sponding dynamical system. (b) If you have access to computer software that is capable of solving event problems, solve for and plot the graphs of 298 / 683 x(1) and x' (1) for the following sets of initial conditions: 1, x2 c 0, x < c. и(х — с) 3D Use the unit step function to express the differential equation in Problem 1 in a single line. i. x(0) = 2, x'(0) = 0 ii. x(0) = 5, x'(0) = 0 3. Is the differential equation derived in Problems 1 and 2 linear or nonlinear? Explain why. Give a physical explanation of why the limiting values of the trajectories as t → o depend on the initial conditions. (c) Draw a phase portrait for the equivalent dynamical sys- 4. In the case that the damping constant y > 0, find the crit- ical points of the differential equation in Problem 2 and dis- cuss their stability properties. 5. Consider the case of an undamped problem (y = 0) using the parameter values L = 1, m = 1, k = 1 and initial condi- tem. 7. Describe some other physical problems that could be for- mulated as event problems in differential equations.
Transcribed Image Text:10:14 PM B Select text O: Select all Copy 1. Assuming that both springs have spring constant k and that there is a damping force proportional to velocity x' with damping constant y, write down the differential equation that describes the motion of the mass. Note that the expression for the force function depends on which of the three different parts of its domain x lies within. tions x(0) = 2, x'(0) = 0. Find the solution x(t) of the initial value problem for 0<1< and si<5, where t and t; are the times of the first and second events. Describe the physical situation corresponding to each of these two events. 6. Consider the damped problem using the parameter values L = 1, m = 1, y = 0.1, and k = 1. 2. The Heaviside, or unit step function, is defined by (a) Use a computer to draw a direction field of the corre- sponding dynamical system. (b) If you have access to computer software that is capable of solving event problems, solve for and plot the graphs of 298 / 683 x(1) and x' (1) for the following sets of initial conditions: 1, x2 c 0, x < c. и(х — с) 3D Use the unit step function to express the differential equation in Problem 1 in a single line. i. x(0) = 2, x'(0) = 0 ii. x(0) = 5, x'(0) = 0 3. Is the differential equation derived in Problems 1 and 2 linear or nonlinear? Explain why. Give a physical explanation of why the limiting values of the trajectories as t → o depend on the initial conditions. (c) Draw a phase portrait for the equivalent dynamical sys- 4. In the case that the damping constant y > 0, find the crit- ical points of the differential equation in Problem 2 and dis- cuss their stability properties. 5. Consider the case of an undamped problem (y = 0) using the parameter values L = 1, m = 1, k = 1 and initial condi- tem. 7. Describe some other physical problems that could be for- mulated as event problems in differential equations.
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