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Consider the equation for a mass m > 0 on a spring with spring-constant k > 0, with damping friction b>0: the damped oscillator mx + bx + kx = 0. (a) Define v(t) = (t) and write this second order equation as a pair of coupled first-order equations for x(t) and v(t). (b) For various values of the parameters, classify all possible solution types, their fixed points and stability. [You should probably divide through by m first..]. Interpret the results physically. = c) Show that if b 0, then v2 + ax² trajectories are ellipses. = constant for some a. What is a? Thus show the

Consider the equation for a mass m > 0 on a spring with spring-constant k > 0, with damping friction b>0: the damped oscillator mx + bx + kx = 0. (a) Define v(t) = (t) and write this second order equation as a pair of coupled first-order equations for x(t) and v(t). (b) For various values of the parameters, classify all possible solution types, their fixed points and stability. [You should probably divide through by m first..]. Interpret the results physically. = c) Show that if b 0, then v2 + ax² trajectories are ellipses. = constant for some a. What is a? Thus show the

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the equation for a mass m > 0 on a spring with spring-constant k > 0, with damping friction b > 0: the damped oscillator mx + bx + kx 0. (a) Define v(t) = x(t) and write this second order equation as a pair of coupled first-order equations for x(t) and v(t). (b) For various values of the parameters, classify all possible solution types, their fixed points and stability. [You should probably divide through by m first..]. Interpret the results physically. - (c) Show that if b 0, then v² + ax² = constant for some a. What is a? Thus show the trajectories are ellipses.
Transcribed Image Text:Consider the equation for a mass m > 0 on a spring with spring-constant k > 0, with damping friction b > 0: the damped oscillator mx + bx + kx 0. (a) Define v(t) = x(t) and write this second order equation as a pair of coupled first-order equations for x(t) and v(t). (b) For various values of the parameters, classify all possible solution types, their fixed points and stability. [You should probably divide through by m first..]. Interpret the results physically. - (c) Show that if b 0, then v² + ax² = constant for some a. What is a? Thus show the trajectories are ellipses.
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