20. Consider the equation d²y d1² +=y=0 m for the motion of a simple harmonic oscillator. (a) Consider the function y(t) = cos ßt. Under what conditions on ß is y(t) a solution? (b) What initial condition (t = 0) in the yv-plane corresponds to this solution? (c) In terms of k and m, what is the period of this solution? (d) Sketch the solution curve (in the yv-plane) associated to this solution. [Hint: Consider the quantity y² + (v/B)².]
20. Consider the equation d²y d1² +=y=0 m for the motion of a simple harmonic oscillator. (a) Consider the function y(t) = cos ßt. Under what conditions on ß is y(t) a solution? (b) What initial condition (t = 0) in the yv-plane corresponds to this solution? (c) In terms of k and m, what is the period of this solution? (d) Sketch the solution curve (in the yv-plane) associated to this solution. [Hint: Consider the quantity y² + (v/B)².]
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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![20. Consider the equation
+
dt² m
for the motion of a simple harmonic oscillator.
0
(a) Consider the function y(t) = cos ßt. Under what conditions on ß is y(t) a
solution?
(b) What initial condition (t = 0) in the yv-plane corresponds to this solution?
(c) In terms of k and m, what is the period of this solution?
(d) Sketch the solution curve (in the yv-plane) associated to this solution. [Hint:
Consider the quantity y² + (v/B)².]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F464fe16a-00d5-4182-8379-61bd4b8bbe02%2F2aa2f210-05f2-44b9-99cf-544c5071b815%2F8c90urs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:20. Consider the equation
+
dt² m
for the motion of a simple harmonic oscillator.
0
(a) Consider the function y(t) = cos ßt. Under what conditions on ß is y(t) a
solution?
(b) What initial condition (t = 0) in the yv-plane corresponds to this solution?
(c) In terms of k and m, what is the period of this solution?
(d) Sketch the solution curve (in the yv-plane) associated to this solution. [Hint:
Consider the quantity y² + (v/B)².]
Expert Solution
Step 1
The given differential equation is .
(a) The given solution is .
To Find: A condition on for which y(t) is a solution to the differential equation.
(b) To Find: the initial condition (t=0) in the y-v plane corresponding to this solution.
(c) To Find: Period of this solution in terms of k and m.
(d) To Sketch: Some solution curves in y-v plane to this solution.
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