MAT2705-01 08F 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). You may use technology to find the roots of polynomials. [No mercy for mistakes here.]
1. 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 , 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
a) Find the general solution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImhGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== of the related homogeneous DE. Is this system underdamped, critically damped or underdamped and why?b) Use the method of undetermined coefficients to find the steady state sinusoidal solution (the particular solution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRInBGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== ) of this driven harmonic oscillator DE.c) Find the solution of the initial value problem.d) Evaluate the amplitude LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and phase shift LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy of the steady state solution function, and numerically evaluate these formulas, and evaluate the fraction \316\264/(2\317\200) to see what fraction of a cycle the phase is shifted. What is the period LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= of the steady state solution?e) Identify the values of the natural decay constant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+, the corresponding characteristic time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5NjQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnRjUvRjNRJXRydWVGJy9GNlEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1, the natural frequency LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnRjUvRjNRJXRydWVGJy9GNlEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GLzYjUSFGJ0Y1, and the quality factor 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.f) Show that the amplitude agrees with the formula
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 with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjJGJ0YyRjI=.g) Optional. Use technology to plot both the solution and the steady state solution together on the same axes as well as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjY2LVEqJnVtaW51czA7RidGOUY7L0Y/Rj1GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORlVGK0Y5 for an appropriate window (say 2 periods of the steady state solution) in which one sees clearly the merging of the two curves as the solution reaches the steady state at least to the pixel size of your technology grapher. Sketch what you see, labeling the axes, tickmarks, etc. Does your steady state solution agree with your amplitude envelope? Does the phase shift fraction of a full cycle look right? Does the time interval it takes for the difference between the two solutions to disappear agree with the characteristic time of the homogeneous solution (called the transient). Explain.solution1.a)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a) b)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 homogeneous solution (terms with arbitrary constants) is a damped oscillation, so it is underdamped. The rest of the solution is the steady state solution.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c)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d)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)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkZGVxRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnQy0+SSNrMEc2IiIiJSIiIj5JJXRhdTBHRiUqJEYkISIiRic+SSdvbWVnYTBHRiUtSSVzcXJ0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMiIilGJy1JJmV2YWxmR0YxNiNJIiVHRiVGJz5JIlFHRiUqJkYtRidGKUYnRidGNQ==f)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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEmZXZhbGZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSJBRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkBGPUZAg)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEmZXZhbGZGJ0YvRjItSShtZmVuY2VkR0YkNiQtRiM2I0YrRjk=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The calculated amplitude (gray horizontal lines) indeed describes the steady state solution (blue).The characteristic decay time for the transient is 1/2 and clearly by 5 such characteristic times, the transient (difference between red and blue curves) has disappeared in the graphic as the full solution curve (red) merges with the steady state solution (blue).The positive phase shift by about 0.17 cycles (roughly 1/6 of a cycle) shifts the graph to the right by about that amount.