Question

Differential equation class. Please show steps to the solutions.

Section 3.3 Exercises To Solutions For all exercises in this section you will be working with the equation dt for various values of m, β and k. but always with f(t)-0. 1. (a) Solve the initial value problem consisting of Equation (1) withm-5, B- and k 80, and initial conditions y(02, y(0)-6. Give your answer in the form y Cesin(wt and all numbers in decimal form, rounded to the nearest tenth. (Note that 5.0007 rounded to the nearest tenth is 5.0, not 5! What is the difference?) (b) Graph the solution to the IVP on your calculator. Adjust the viewing window to get about three cycles of the motion displayed fairly large. Sketch your graph. (c) Graph y 2.3e.and y -2.3e0.6 together with the solution you graphed in (b) Add them to your sketch as dashed curves. What you have just seen is an example of what is called underdamped vibration. There is damping but it is smal enough to allow the mass to move up and down while the vibration decays. 2, (a) Solve the initial value problem consisting of Equation (1) with m-5, β-50 and k-80. and initial conditions y(0)2, '(0)6. (b) Sketch the graph of your solution over a time period long enough to show what is happening over time. Make your vertical scale such that the general shape of the graph can be seen. (c) Compare and contrast your solution to this IVP with the solution to the IVP from Exercise 1, in terms of amplitude over time and oscillation. For your answer to 2(c) you should have noted that the solution of the IVP in Exercise 1 oscillates as it decays, and the solution of the IVP in Exercise 2 does not. We say that the situation from Exercise 2 is overdamped vibration. The initial conditions do not affect whether the vibration will be underdamped or overdamped (can you see why not?), so the only difference is the value of B 3. (a) Determine a value of B that is the "dividing line" between equations have solutions that oscillate as they decay and those that do not oscillate. Explain how you determined it. (b) Solve the IVP with m = 5, k 80 and the value of β you obtained in (a), along with the initial conditions from Exercise 2. Graph the solution using some technology and sketch the graph. (c Suppose now that m and 80. How would you expect the solution to (1) to behave if was slightly smaller than the value obtained in (a) and why. How would you expect the solution to behave for slightly larger than the value obtained in (a)? Answer in complete sentences. these For your answer to 3(c) you should have said that if β is a little less than the value that you found in (a) the solution will behave like that in Exercise 1, and if B is a little more than that value the solution will behave like that in Exercise 2. The situation in Exercises 3(a) and 3(b) is called the critically damped case, meaning that if there is just a little less damping the behaviorl be oscillatory, but if the damping has that critical value or greater, the motion will not be oscillatory. 4. Describe a method for determining from the coefficients m, β and k whether the solution will be underdamped, overdamped, or critically damped.

Answer 1

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