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In Exercises 13-20, consider harmonic oscillators with mass m, spring constant k, and damping coefficient b. For the values specified, (a) write the second-order differential equation and the corresponding first-order sys tem; (b) find the eigenvalues and eigenvectors of the linear system; (e) classify the oscillator (as underdamped, overdamped, critically damped, or un- damped) and, when appropriate, give the natural period; (d) sketch the phase portrait of the associated linear system and include the solution curve for the given initial condition; and (e) sketch the y(t)- and u(t)-graphs of the solution with the given initial condition. 13. m = 1, k = 7, b = 8, with initial conditions y(0) = -1, v(0) = 5 14. m= 1, k-8, b = 6, with initial conditions y(0) = 1, v(0) = 0 15. m = 1, k = 5, b = 4, with initial conditions y(0) = 1, v(0) = 0 16. m = 1, k-8, b = 0, with initial conditions y(0) = 1, v(0) = 4 17. m = 2, k = 1, b = 3, with initial conditions y(0) = 0. v(0) = 3 18. m = 9, k = 1, b = 6, with initial conditions y(0) = 1, v(0) = 1 19. m = 2, k = 3, b= 0, with initial conditions y(0) = 2, v(0) = -3 20. m2, k = 3, b= 1, with initial conditions y(0)= 0, v(0) = -3

In Exercises 13-20, consider harmonic oscillators with mass m, spring constant k, and damping coefficient b. For the values specified, (a) write the second-order differential equation and the corresponding first-order sys tem; (b) find the eigenvalues and eigenvectors of the linear system; (e) classify the oscillator (as underdamped, overdamped, critically damped, or un- damped) and, when appropriate, give the natural period; (d) sketch the phase portrait of the associated linear system and include the solution curve for the given initial condition; and (e) sketch the y(t)- and u(t)-graphs of the solution with the given initial condition. 13. m = 1, k = 7, b = 8, with initial conditions y(0) = -1, v(0) = 5 14. m= 1, k-8, b = 6, with initial conditions y(0) = 1, v(0) = 0 15. m = 1, k = 5, b = 4, with initial conditions y(0) = 1, v(0) = 0 16. m = 1, k-8, b = 0, with initial conditions y(0) = 1, v(0) = 4 17. m = 2, k = 1, b = 3, with initial conditions y(0) = 0. v(0) = 3 18. m = 9, k = 1, b = 6, with initial conditions y(0) = 1, v(0) = 1 19. m = 2, k = 3, b= 0, with initial conditions y(0) = 2, v(0) = -3 20. m2, k = 3, b= 1, with initial conditions y(0)= 0, v(0) = -3

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 63EQ
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HELP PLEASE!!! 17 & 18 Please!

In Exercises 13-20, consider harmonic oscillators with mass m, spring constant k, and damping coefficient b. For the values specified, (a) write the second-order differential equation and the corresponding first-order sys- tem; (b) find the eigenvalues and eigenvectors of the linear system; (c) classify the oscillator (as underdamped, overdamped, critically damped, or un- damped) and, when appropriate, give the natural period; (d) sketch the phase portrait of the associated linear system and include the solution curve for the given initial condition; and (e) sketch the y(t)- and v(t)-graphs of the solution with the given initial condition. 13. m = 1, k = 7, b = 8, with initial conditions y(0) = -1, v(0) = 5 14. m = 1, k = 8, b = 6, with initial conditions y(0) = 1, v(0) = 0 15. m = 1, k = 5, b = 4, with initial conditions y(0) = 1, v(0) = 0 16. m = 1, k = 8, b = 0, with initial conditions y(0) = 1, v(0) = 4 17. m = 2, k = 1, b = 3, with initial conditions y(0) = 0, v(0) = 3 18. m = 9, k = 1, b = 6, with initial conditions y(0) = 1, v(0) = 1 19. m = 2, k = 3, b = 0, with initial conditions y(0) = 2, v(0) = -3 20. m = 2, k = 3, b = 1, with initial conditions y(0) = 0, v(0) = -3
Transcribed Image Text:In Exercises 13-20, consider harmonic oscillators with mass m, spring constant k, and damping coefficient b. For the values specified, (a) write the second-order differential equation and the corresponding first-order sys- tem; (b) find the eigenvalues and eigenvectors of the linear system; (c) classify the oscillator (as underdamped, overdamped, critically damped, or un- damped) and, when appropriate, give the natural period; (d) sketch the phase portrait of the associated linear system and include the solution curve for the given initial condition; and (e) sketch the y(t)- and v(t)-graphs of the solution with the given initial condition. 13. m = 1, k = 7, b = 8, with initial conditions y(0) = -1, v(0) = 5 14. m = 1, k = 8, b = 6, with initial conditions y(0) = 1, v(0) = 0 15. m = 1, k = 5, b = 4, with initial conditions y(0) = 1, v(0) = 0 16. m = 1, k = 8, b = 0, with initial conditions y(0) = 1, v(0) = 4 17. m = 2, k = 1, b = 3, with initial conditions y(0) = 0, v(0) = 3 18. m = 9, k = 1, b = 6, with initial conditions y(0) = 1, v(0) = 1 19. m = 2, k = 3, b = 0, with initial conditions y(0) = 2, v(0) = -3 20. m = 2, k = 3, b = 1, with initial conditions y(0) = 0, v(0) = -3
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