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8. Consider any damped harmonic oscillator equation d² y dt2 m d²y dt² dy dt +b. + ky = 0. (a) Show that the sum of any two solutions is another solution. (b) Using the result of part (a), solve the initial-value problem dy +3 + 2y = 0, y (0) = 2, v(0) = -3. dt (c) Using the result of part (a) in Exercise 7 along with the result of part (a) of this exercise, solve the initial-value problem dy TH nairU (x). d² y +3. + 2y = 0, y(0) = 3, v(0) = -5. dt² dt (d) How many solutions to the equation d² y dy dt² dt +3. + 2y = 0 do you get if you use the results of Exercise 7 and this exercise along with the guess-and-test method described in this section?

8. Consider any damped harmonic oscillator equation d² y dt2 m d²y dt² dy dt +b. + ky = 0. (a) Show that the sum of any two solutions is another solution. (b) Using the result of part (a), solve the initial-value problem dy +3 + 2y = 0, y (0) = 2, v(0) = -3. dt (c) Using the result of part (a) in Exercise 7 along with the result of part (a) of this exercise, solve the initial-value problem dy TH nairU (x). d² y +3. + 2y = 0, y(0) = 3, v(0) = -5. dt² dt (d) How many solutions to the equation d² y dy dt² dt +3. + 2y = 0 do you get if you use the results of Exercise 7 and this exercise along with the guess-and-test method described in this section?

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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8. Consider any damped harmonic oscillator equation d²y dt² m dy +b. dt (a) Show that the sum of any two solutions is another solution. (b) Using the result of part (a), solve the initial-value problem dy +3. + 2y = 0, y(0) = 2, v(0) = - dt -3. +3 + ky = 0. dt² (c) Using the result of part (a) in Exercise 7 along with the result of part (a) of this exercise, solve the initial-value problem d²y dy deve dt² dt (d) How many solutions to the equation +2y = 0, nie! (n) y(0) = 3, v(0) = -5. d²y dy dt² dt +3. + 2y = 0 do you get if you use the results of Exercise 7 and this exercise along with the guess-and-test method described in this section?
Transcribed Image Text:8. Consider any damped harmonic oscillator equation d²y dt² m dy +b. dt (a) Show that the sum of any two solutions is another solution. (b) Using the result of part (a), solve the initial-value problem dy +3. + 2y = 0, y(0) = 2, v(0) = - dt -3. +3 + ky = 0. dt² (c) Using the result of part (a) in Exercise 7 along with the result of part (a) of this exercise, solve the initial-value problem d²y dy deve dt² dt (d) How many solutions to the equation +2y = 0, nie! (n) y(0) = 3, v(0) = -5. d²y dy dt² dt +3. + 2y = 0 do you get if you use the results of Exercise 7 and this exercise along with the guess-and-test method described in this section?
7. Consider any damped harmonic oscillator equation d²y dy dt² +b dt m + ky = 0. (a) Show that a constant multiple of any solution is another solution. (b) Illustrate this fact using the equation d² y dt² dy +3+2y=0 dt discussed in the section. (c) How many solutions to the equation do you get if you use this observation along with the guess-and-test method described in this section?
Transcribed Image Text:7. Consider any damped harmonic oscillator equation d²y dy dt² +b dt m + ky = 0. (a) Show that a constant multiple of any solution is another solution. (b) Illustrate this fact using the equation d² y dt² dy +3+2y=0 dt discussed in the section. (c) How many solutions to the equation do you get if you use this observation along with the guess-and-test method described in this section?
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