MAT2705-02/05 15S Quiz 8 Print Name (Last, First) _________________________________|__Show all work, including mental steps, in a clearly organized way that speaks for itself. Use proper mathematical notation, identifying expressions by their proper symbols (introducing them if necessary), and use equal signs and arrows when appropriate. Always SIMPLIFY expressions. BOX final short answers. LABEL parts of problem. Keep answers EXACT (but give decimal approximations for interpretation). Indicate where technology is used and what type (Maple, GC).1. m 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 , 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, y '(0) = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkMtSSNtbkdGJDYkUSIxRidGL0Yv; m = 4, c = 4, k = 37 . [prime is d/dtLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LlEiXUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR1EmZmFsc2VGJy8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSwwLjE2NjY2NjdlbUYnLyUncnNwYWNlR1EsMC4xMTExMTExZW1GJ0YvRjI=
a) State Maple's solution of the initial value problem (use function notation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tRiw2I1EhRidGPQ==).b) Put the DE into standard linear form first. Then identify the values of the damping constant and characteristic time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdGPS1JI21vR0YkNi1RIj1GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlctRjs2JFEiMUYnRj4tRkM2LVEiL0YnRj5GRkZJL0ZMRjRGTUZPRlFGUy9GVlEsMC4xNjY2NjY3ZW1GJy9GWUZcby1GLDYlLUYvNiVRJyYjOTY0O0YnL0YzRkhGPkY4RkBGPg== , the natural frequency LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnRjUvRjNRJXRydWVGJy9GNlEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdGPS1GLzYjUSFGJ0Y1 , and the quality factor 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LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEmJnRhdTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUklbXJvd0dGJDYkLUkjbW5HRiQ2JFEiMEYnRjJGMi8lL3N1YnNjcmlwdHNoaWZ0R0Y7, exactly and numerically. Is this underdamped, critically damped or overdamped?
c) Find the general solution by hand, showing all steps.d) Find the solution satisfying the initial conditions, showing all steps.
e) Give exact and numerical values of the amplitude and phase shift and re-express the sinusoidal factor of this solution in phase-shifted cosine form. [Make sure you use a diagram to justify your values.] State what numerical fraction of a cycle (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkctSSNtaUdGJDYlUScmIzk2MDtGJy8lJ2l0YWxpY0dGOEYvRi8=) the phase shift is (i.e., evaluate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy/2\317\200) as well as its numerical value in degrees, and whether the cosine curve is shifted left (earlier in time) or right (later in time) on the time line (by a phase less than or equal to half a cycle of course). Explain.
f) State the two envelope functions of this decaying oscillating solution.g) Make a rough sketch of the plot of your solution and its two envelope functions in a viewing window of width 5 times the characteristic time of the solution exponential factor. h) What are the numerical values of the periods associated with the natural frequency and the actual frequency of the sinusoidal factor of the solution and how do they compare? solution1.a) 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 so 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. very underdamped.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The solution frequency of 2 is just a bit below the natural frequency.This system is way underdamped, comparing: 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c)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The general solution is a linear combination of the real and imaginary parts of this complex exponential.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d)details of initial conditionsPkkiWUc2ImYqNiNJInRHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJComLUkkZXhwR0YkNiMsJDkkIyEiIiIiIyIiIiwmKiZJI2MxR0YkRjQtSSRjb3NHRiQ2IywkRjAiIiRGNEY0KiZJI2MyR0YkRjQtSSRzaW5HRiRGOkY0RjRGNEYkRiRGJA==We can work backwards to get integer solutions:QyU2JC8tSSJZRzYiNiMiIiFJI3kwR0YnLy1JJWV2YWxHJSpwcm90ZWN0ZWRHNiQtSSVkaWZmR0YuNiQtRiY2I0kieEdGJ0Y1L0Y1RilJI3YwR0YnIiIiLUkmc29sdmVHRic2JDwjSSIlR0YnPCRJI2MxR0YnSSNjMkdGJw==QyU2JC8tSSJZRzYiNiMiIiEhIiMvLUklZXZhbEclKnByb3RlY3RlZEc2JC1JJWRpZmZHRi42JC1GJjYjSSJ4R0YnRjUvRjVGKSEiIiIiIi1JJnNvbHZlRzYkRi5JKF9zeXNsaWJHRic2JDwjSSIlR0YnPCRJI2MxR0YnSSNjMkdGJw==JSFHe) The coefficient pair is in the second quadrant.QyQtSSV3aXRoRzYiNiNJJnBsb3RzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlISIiLUkoZGlzcGxheUc2IjYkLUkmYXJyb3dHRiQ2Ji1JJDwsPkdJKF9zeXNsaWJHRiQ2JCEiIyNGLSIiJC9JJnNoYXBlR0YkRicvSSZjb2xvckdGJEkkcmVkR0YkL0kqdGhpY2tuZXNzR0YkIiIjL0koc2NhbGluZ0dGJEksY29uc3RyYWluZWRHRiQ=QyU+SSJBRzYiLUklc3FydEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCYtSSJeR0YpNiQhIiMiIiMiIiItRi42JCNGMCIiJEYxRjJGMi1JJmV2YWxmR0YpNiNJIiVHRiU=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So the peaks are shifted left with respect to a standard cosine wave, by a bit less than one half a cycle, apparent below.Here is a check that this indeed represents our original expression:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY0LUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUY2Ni1RIidGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMTExMTExMWVtRicvRk5RJjAuMGVtRidGK0ZPLUY2Ni1RIjpGJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUYsNiVRIkFGJ0YvRjItRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSC9GS0ZVRlQtRiw2JVEkY29zRicvRjBGPUY5LUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JUYuRl1vRjktRjY2LVEoJm1pbnVzO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZpby1GLDYlUSgmZGVsdGE7RidGXW9GOUY5LUY2Ni1RIjtGJ0Y5RjsvRj9GMUZARkJGREZGRkhGaW5GTUZmbi1GLDYlUSdleHBhbmRGJ0YvRjItRl9vNiQtRiM2Iy1GLDYlUSIlRidGL0YyRjlGXnBGZm4tRiw2JVElc3Vic0YnRi9GMi1GX282JC1GIzYpRistRjY2LVEiPUYnRjlGO0Y+RkBGQkZERkZGSEZKRk0tSSNtbkdGJDYkUSIzRidGOUZmbi1GLDYlUSJ0RidGL0YyLUY2Ni1RIixGJ0Y5RjtGYXBGQEZCRkRGRkZIRmluL0ZOUSwwLjMzMzMzMzNlbUYnRmlwRjk=a bit more than 1/3 of a cycle to the left, so the first peak (green) is ahead in time compared to the standard cosine (blue)., but to the naked eye here, it seems about exactly half a cycle out of phase. Compare the damped red curve with the undamped green curve.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)
The envelope functions 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)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 solution period is just a big longer than the natural period (damping slows down the system), but very close due to the high LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= value.If we examine resonance for this system, you see the nice resonance peak at 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