MAT2705-01/02 12F 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 arrows and equal signs 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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 , y(0) =0, y '(0) = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkMtSSNtbkdGJDYkUSI4RidGL0Yv; m = 2, c = 12, k = 50 . [prime is d/dtLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LlEiXUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR1EmZmFsc2VGJy8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSwwLjE2NjY2NjdlbUYnLyUncnNwYWNlR1EsMC4xMTExMTExZW1GJ0YvRjI=
a) Put the DE into standard linear form first. Then identify the values of the damping constant 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 , the natural frequency LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnRjUvRjNRJXRydWVGJy9GNlEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdGPS1GLzYjUSFGJ0Y1 , and the quality factor 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LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEmJnRhdTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUklbXJvd0dGJDYkLUkjbW5HRiQ2JFEiMEYnRjJGMi8lL3N1YnNjcmlwdHNoaWZ0R0Y7. Is this underdamped, critically damped or overdamped?
b) Find the general solution by hand.c) Find the solution satisfying the initial conditions.
d) Re-express the sinusoidal factor of this solution in phase-shifted cosine form to obtain the two envelope functions of this decaying oscillation solution. State the two envelope functions, and state what fraction of a cycle (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkctSSNtaUdGJDYlUScmIzk2MDtGJy8lJ2l0YWxpY0dGOEYvRi8=) the phase shift is and whether the cosine is shifted left (earlier in time) or right (later in time) on the time line.d) 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.e) State Maple's solution of the initial value problem. solution1.a) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==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Observe 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 so 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. slightly underdamped.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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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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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 is a check that this indeed represents our original expression: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1/4 of a cycle to the left, so the first peak (green) is ahead in time compared to the standard cosine (blue). Compare the damped red curve with the undamped green curve.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QyUvSSJURzYiLUkiKkclKnByb3RlY3RlZEc2JCwkSSNQaUdGKCIiIyMiIiIiIiVGLi1JJmV2YWxmR0YoNiNJIiVHRiU=JSFH