MAT2705-01/02 13F Take Home Test 3 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, MathCad). You may use technology for row reductions, matrix inverses, plotting and root finding without showing intermediate steps. Print the requested 8 technology plots, labeling them and annotating them appropriately by hand and attach to the end of your test. All differential equations should be solved "by hand".
1. The displacement LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== of an underdamped harmonic oscillator system satisfies
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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSIuRidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZURj0=
Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjJGJ0Y5Rjk= , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjhGJ0Y5Rjk= , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRJDEzNkYnRjlGOQ== and the initial conditions 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 .
a) Express this DE in standard form with a unit coefficient of the second derivative term. What are the natural frequency LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JJW1yb3dHRiQ2JC1JI21uR0YkNiRRIjBGJ0YyRjIvJS9zdWJzY3JpcHRzaGlmdEdGOw== , natural decay time LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXJvd0dGJDYkLUkjbWlHRiQ2JVEmJnRhdTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjUtRiw2JC1JI21uR0YkNiRRIjBGJ0Y1RjUvJS9zdWJzY3JpcHRzaGlmdEdGPQ== , the quality factor 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and the period 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/LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJm9tZWdhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSVtcm93R0YkNiQtSSNtbkdGJDYkUSIwRidGMkYyLyUvc3Vic2NyaXB0c2hpZnRHRjs= for this system? (Give both exact and numeric values to 3 decimal places.)
Consider the following driving force functions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== :
b) 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. Find the general solution of the differential equation. What is the frequency LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUYjNiQtRiw2JS1JI21pR0YkNiVRJyYjOTY5O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjFGJ0Y5L0Y3USV0cnVlRicvRjpRJ2l0YWxpY0YnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGOS1GIzYkLUYzNiNRIUYnRjlGRkY5 , decay time LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXJvd0dGJDYkLUkjbWlHRiQ2JVEmJnRhdTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjUtRiw2JC1JI21uR0YkNiRRIjFGJ0Y1RjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== and the period LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIj1GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlgtRjs2JFEiMkYnRj4tRkQ2LVEifkYnRj5GR0ZKRkxGTkZQRlJGVC9GV1EmMC4wZW1GJy9GWkZcby1GLzYlUScmIzk2MDtGJy9GM0ZJRj5GPg==/LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJm9tZWdhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSVtcm93R0YkNiQtSSNtbkdGJDYkUSIxRidGMkYyLyUvc3Vic2NyaXB0c2hpZnRHUSIwRic= for this decaying sinusoidal solution? (Give both exact and numeric values to 3 decimal places.c) 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. Solve the above initial value problem for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHRj0tSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZXLUY7NiRRIjFGJ0Y+Rj4=. Evaluate the two envelope functions for the exponentially decaying sinusoidal function and plot (#1) all three together for 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LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXJvd0dGJDYkLUkjbWlHRiQ2JVEmJnRhdTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjUtRiw2JC1JI21uR0YkNiRRIjFGJ0Y1RjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw==, fully labeling the axes and three curves. [You should see a number of oscillations before convergence to the horizontal axis.]
d) 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. Find the initial value problem solution for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHRj0tSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZXRjpGPg== by hand (but check with Maple!).e) Evaluate the values of the amplitude A and phase shift \316\264 for the steady state solution (the part of the solution which remains after the transient has died away). Recall that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbWlHRiQ2JVEkY29zRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRJyYjOTY5O0YnRi9GMi1JI21vR0YkNi1RIn5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJ0RicvRjBRJXRydWVGJy9GM1EnaXRhbGljRidGMkYyLUY+Ni1RIitGJ0YyRkFGQ0ZFRkdGSUZLRk0vRlBRLDAuMjIyMjIyMmVtRicvRlNGaW4tRiw2JVEkc2luRidGL0YyRjUtRj42LVEiPUYnRjJGQUZDRkVGR0ZJRktGTS9GUFEsMC4yNzc3Nzc4ZW1GJy9GU0Ziby1GIzYlLUYsNiNRIUYnLUkmbXNxcnRHRiQ2Iy1GIzYkLUkjbW5HRiQ2JFEiMkYnRjJGMkYyRistRjY2JC1GIzYqRjpGPUZULUY+Ni1RKCZtaW51cztGJ0YyRkFGQ0ZFRkdGSUZLRk1GaG5Gam4tRiw2JVEnJiM5NjA7RidGL0YyLUY+Ni1RIi9GJ0YyRkFGQy9GRkZYRkdGSUZLRk0vRlBRLDAuMTY2NjY2N2VtRicvRlNGYXEtRl9wNiRRIjRGJ0YyRjJGMkYy, so the driving function has phase shift 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. The phase shift of the solution relative to the driving function is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RKCZtaW51cztGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGSS1JJW1zdWJHRiQ2JUYrLUYjNiUtRiw2JVEic0YnL0YwUSV0cnVlRicvRjNRJ2l0YWxpY0YnRlRGVi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjI= , i.e., the phase by which the graph is shifted to the right on graph versus time compared to the sine function. Express this relative phase shift in radians, degrees and cycles (divide radians by 2\317\200).
f) Make a single plot (#2) in an appropriate viewing window (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) showing both the solution function and the steady state solution until they merge. In a separate plot for comparison with the driving sine function, plot (#3) both the steady state solution and 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 (same amplitude as the steady state solution) to see how the peaks of the steady state solution compare to the peaks of the driving function. Does your plot agree with your calculated relative phase shift angle LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RKCZtaW51cztGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGSS1JJW1zdWJHRiQ2JUYrLUYjNiUtRiw2JVEic0YnL0YwUSV0cnVlRicvRjNRJ2l0YWxpY0YnRlRGVi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjI= (does the steady state solution lead ahead in time [peaks earlier] or lag behind in time [peaks later] the driving function by a corresponding amount)? Explain.g) 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 .Explore resonance for this system by finding the steady state solution by hand, where the frequency LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy of the driving force function is a parameter. h) Evaluate the steady state amplitude function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRJyYjOTY5O0YnL0YwUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0Y/Rj8tSSNtb0dGJDYtUSJ+RidGPy8lJmZlbmNlR0Y+LyUqc2VwYXJhdG9yR0Y+LyUpc3RyZXRjaHlHRj4vJSpzeW1tZXRyaWNHRj4vJShsYXJnZW9wR0Y+LyUubW92YWJsZWxpbWl0c0dGPi8lJ2FjY2VudEdGPi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlVGPw==and use calculus to find the exact and numerical values of both the frequency LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiM2JC1GLDYlUSgmb21lZ2E7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y6LUYjNiQtRiw2JVEicEYnL0Y4USV0cnVlRicvRjtRJ2l0YWxpY0YnRjovJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0YrRjo= and the amplitude LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUklbXN1YkdGJDYlLUYsNiVRKCZvbWVnYTtGJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRictRiM2JC1GLDYlUSJwRidGL0YyRkIvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZCRkJGQg== where it has its peak value for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RLyZHcmVhdGVyRXF1YWw7RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkktSSNtbkdGJDYkUSIwRidGMkYy. Does LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiOEYnL0YzUSdub3JtYWxGJ0Y+Rj5GPg== with 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 agree with your value for A from part d) as it should? What is the numerical value 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 for 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? How does the value of 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/LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMEYnL0YzUSdub3JtYWxGJ0Y+Rj5GPg== compare to the quality factor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ? i) Set LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1JI21uR0YkNiRRIjBGJy9GM1Enbm9ybWFsRidGPC8lL3N1YnNjcmlwdHNoaWZ0R0Y7=130 and plot (#4) the amplitude function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRJyYjOTY5O0YnL0YwUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0Y/Rj8tRiw2I1EhRidGPw== in an appropriate window (showing the limiting behavior of the entire function for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RLyZHcmVhdGVyRXF1YWw7RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkktSSNtbkdGJDYkUSIwRidGMkYy) together with the constant functions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiNRIUYnLUYjNiZGOi1JJW1zdWJHRiQ2JS1GLDYlUScmIzk2OTtGJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRictRiM2JC1GLDYlUSJwRidGL0YyRkcvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y6RkdGOkZHRkdGOkZH (its peak value) and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiNRIUYnLUYjNiZGOi1JJW1zdWJHRiQ2JS1GLDYlUScmIzk2OTtGJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRictRiM2JC1JI21uR0YkNiRRIjBGJ0ZHRkcvJS9zdWJzY3JpcHRzaGlmdEdGTkY6RkdGOkZHRkdGOkZH and hand annotate on your axes the values of these frequencies and amplitudes and indicate the points on the curve which correspond to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JS1JJW1zdWJHRiQ2JS1GLDYlUSgmb21lZ2E7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnRjpGOi8lL3N1YnNjcmlwdHNoaWZ0R0ZCRitGOkYrRjo= and LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJm9tZWdhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSVtcm93R0YkNiQtRiw2JVEicEYnL0YwUSV0cnVlRicvRjNRJ2l0YWxpY0YnRjIvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw==.2. 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,
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 ; LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImlGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSI9RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlctRi82JVEickYnRjJGNS1GQTYtUSIvRidGREZGRkkvRkxGNEZNRk9GUUZTL0ZWUSwwLjE2NjY2NjdlbUYnL0ZZRlxvLUYsNiUtRi82JVEiVkYnRjJGNUY4Rj1GRA== , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjZGJ0Y5Rjk= , 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 .a) Write this closed 3 tank mixing tank system of differential equations for the vector variable 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 = 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 AND its initial conditions in explicit matrix form (showing all components, not just vector symbols) and identifying the coefficient matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.b) Use the eigenvector approach to find its general solution by hand, showing all steps.c) Find the IVP solution, using matrix methods showing all steps. Make sure it agrees with Maple's solution.d) Evaluate the asymptotic solution 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 for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZzcmFycjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmaW5maW47RidGOUY7Rj5GQEZCRkRGRkZIRkpGTUY5, which is the equilibrium solution approached by your solution.e) Plot (#5) the three solution curves together with their horizontal asymptotes in an appropriate viewing window (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RLyZHcmVhdGVyRXF1YWw7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSIwRidGOS1GNjYtUSIhRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnL0ZORldGOQ==) that shows them reaching and settling down to those equilibrium values without compressing the interesting part before reaching that limit. Label each curve. Comment about your choice of viewing window in relation to the characteristic decay time of the problem.3. 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.a) Identify the coefficient matrix and by hand find a new basis for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0Y+LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y+ consisting of eigenvectors 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 of this matrix using the standard hand recipe, rescaling them to be the minimal integer eigenvectors. Order the real eigenvalues 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 by decreasing value (increasing absolute value). Identify 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.b) Evaluate the new coordinates LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiQtSSNtbkdGJDYkUSIxRicvRjtRJ25vcm1hbEYnRkMvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIixGJ0ZDLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y5LyUpc3RyZXRjaHlHRk4vJSpzeW1tZXRyaWNHRk4vJShsYXJnZW9wR0ZOLyUubW92YWJsZWxpbWl0c0dGTi8lJ2FjY2VudEdGTi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYxNiVGMy1GIzYkLUZANiRRIjJGJ0ZDRkNGRUZDRkMvJSVvcGVuR1EnJmxhbmc7RicvJSZjbG9zZUdRJyZyYW5nO0YnRkM= of the point 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 with respect to this basis of eigenvectors.c) Solve this initial value problem by hand, showing all steps.d) Use technology to plot (#6) a directionfield for this DE with the solution curve through the single initial data point, and (by hand if necessary) include the lines through the two eigenvectors representing the two subspaces of eigenvectors. Choose the window 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. By hand label these lines by their new coordinate labels, draw in and label the eigenvectors and the initial data vector 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 themselves as arrows, and include the parallelogram projection of the latter vector onto the new coordinate axes, i.e., the parallelogram parallel to the new coordinate axes with the initial data vector as the main diagonal. Do the projections along the coordinate axes agree with the values you found for the new coordinates of this vector? Explain. Does your directionfield correspond to the eigenvectors you have drawn? Explain why.e) Plot (#7) the two variables versus LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= for an appropriate viewing window for this initial value problem. Explain your window choice.The sum function 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 has a global maximum (for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=) , obvious in a separate plot (#8) of this variable. Find its coordinates and the corresponding value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= exactly and approximately (Maple is helpful to evaluate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEiKEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EldHJ1ZUYnLyUqc2VwYXJhdG9yR1EmZmFsc2VGJy8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjE2NjY2NjdlbUYnLyUncnNwYWNlR0ZELUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR0Y0L0YwUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnRi9GLy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRi8=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEiK0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZDLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnRi9GTUZQLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGLw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEiKUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EldHJ1ZUYnLyUqc2VwYXJhdG9yR1EmZmFsc2VGJy8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjE2NjY2NjdlbUYnLyUncnNwYWNlR0ZELUkobWZlbmNlZEdGJDYkLUYjNiQtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJ0RicvJSdpdGFsaWNHRjQvRjBRJ2l0YWxpY0YnLUYjNiUtRlA2JVEkbWF4RicvRlRGN0YvRlNGVS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRi9GL0Yv exactly). Annotate your diagram to show this point.Advice. When in doubt about how much work to show, show more. Explain using words if it helps. Think of this take-home test as an exercise in "writing intensive" technical expression. Try to impress bob as though it were material for a job interview. In a real world technical job, you need to be able to write coherent technical reports that other people can follow.solution1. damped harmonic oscillatorLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Iy1GLDYlUShyZXN0YXJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFHPkkmaW5pdHNHNiI2JC8tSSJ4R0YkNiMiIiEiIiIvLS1JIkRHRiQ2I0YoRilGKg==Qy8+SSRkZXFHNiIvLCgqJkkibUdGJSIiIi0tLUkjQEBHRiU2JEkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJSIiIzYjSSJ4R0YlNiNJInRHRiVGKkYqKiZJImNHRiVGKi0tRjBGNUY3RipGKiomSSJLR0YlRiotRjZGN0YqRipJIkZHRiVGKi1JJ2V4cGFuZEdGMjYjKiZGJEYqRikhIiJGKj5GKUY0RkU+RjoiIilGRT5GPiIkTyJGRUYkRipGRA==a)QzE+Jkkmb21lZ2FHNiI2IyIiIS1JJXNxcnRHRiY2IyomSSJLR0YmIiIiSSJtR0YmISIiRi4tSSZldmFsZkclKnByb3RlY3RlZEc2I0kiJUdGJkYuPiZJJHRhdUdGJkYnKiZGL0YuSSJjR0YmRjBGLkYxRi4+SSJRR0YmKiZGJEYuRjdGLkYuLUYyNiRGNSIiJ0YuPiZJIlRHRiZGJywkKiZJI1BpR0YzRi5GJEYwIiIjRi4tRjI2JEY1IiImJSFHb) no forceQyU+SSJGRzYiIiIhISIiKiZJJGRlcUdGJSIiIkkibUdGJUYnLUknZHNvbHZlRzYiNiRJJGRlcUdGJC1JInhHRiQ2I0kidEdGJA==These come from the complex exponentials with rate coefficients given by the roots of this quadratic equation, obtained by substituting the trial exponential 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into the homogeneous DE.LywoKiRJInJHNiIiIiNGJ0YlIiIpIiRPIiIiIiIiIQ==LUkmc29sdmVHNiI2IzwjLywoKiRJInJHRiQiIiNGK0YqIiIpIiRPIiIiIiIiIQ==Qyc+SSR0YXVHNiIjIiIiIiIjISIiPkkiVEdGJS1JIipHJSpwcm90ZWN0ZWRHNiQsJEkjUGlHRi5GKCNGJyIiKUYpLUkmZXZhbGZHRi42I0kiJUdGJQ==QyUtSSZldmFsZkclKnByb3RlY3RlZEc2IywkKiZJJHRhdUc2IiIiIkkiVEdGKiEiIiIiJUYrLUYkNiMsJComSSJRR0YqRitJI1BpR0YlRi1GLg==The characteristic time is 1/2 so 5 characteristic times is 5/2, we go to 3 to see the pixels at the peeks merge completely. We should see about the following number of oscillations in a decay window of width 5 characteristic timesL0kiTkc2Ii1JJmV2YWxmRyUqcHJvdGVjdGVkRzYjLCQqJkkkdGF1R0YkIiIiSSJUR0YkISIiIiImPerhaps 4 decay times is more realistic since the tail is pretty compressed between 4 and 5:QyUtSSZldmFsZkclKnByb3RlY3RlZEc2IywkKiZJJHRhdUc2IiIiIkkiVEdGKiEiIiIiJUYrLUYkNiMsJComSSJRR0YqRitJI1BpR0YlRi1GLg==These are comparable as they should be for this value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEiLkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDRi8=c)QyktSSdkc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JDwlSSRkZXFHRigvLUkieEdGKDYjIiIhIiIiLy0tSSJER0YlNiNGLkYvRjAtRi42I0kidEdGKEYxPkkjeDFHRigtSSRyaHNHRiY2I0kiJUdGKCEiIj5JI0ExR0YoLUklc3FydEdGJTYjLCZGMUYxLUkiXkdGJjYkI0YxIiIlIiIjRjFGMS1JJmV2YWxmR0YmRj4=LUklcGxvdEc2IjYlNyVJI3gxR0YkKiZJI0ExR0YkIiIiLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMsJEkidEdGJCEiI0YqLCRGKCEiIi9GMjsiIiEjIiImIiIjL0kmY29sb3JHRiQ3JUkkcmVkR0YkSSVncmF5R0YkRkA=JSFHd)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 the merging of pixels after about 4 or 5 times the characteristic time.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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 coefficients are 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The difference from 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 response is about 83 degrees behind in time of the driving function curve, just short of a quarter cycle behind (to the right of) that driving function.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Now we plot the steady state solution (blue) and a rescaled driving function (green) to compare the phase 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 blue steady state curve is indeed about 1/4 cycle behind the green driving function curve in time (to the right).Method of undetermined coefficients:SSRkZXFHNiI=QyUtSSVzdWJzRyUqcHJvdGVjdGVkRzYlL0kjRjBHNiIiJEkiL0kieEdGKS1JKHVuYXBwbHlHRik2JCwmKiZJI2MxR0YpIiIiLUkkY29zRzYkRiVJKF9zeXNsaWJHRik2IywkSSJ0R0YpIiIpRjNGMyomSSNjMkdGKUYzLUkkc2luR0Y2RjhGM0YzRjpJJGRlcUdGKSEiIi1JKXNpbXBsaWZ5R0Y2NiNJIiVHRik=LUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JDwkLywmSSNjMUdGJyEja0kjYzJHRiciIikiJEkiLywmRixGL0YuIiNrRjA8JEYsRi4=matrix form of solution:Ly1JMGRlbGF5RG90UHJvZHVjdEc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nRzYkRiZJKF9zeXNsaWJHRik2JS1JJ01hdHJpeEdGKzYjL0kkJWlkR0YpIjV5dUd6VjJXblc9LSZJJ1ZlY3RvckdGKzYjSSdjb2x1bW5HRik2Iy9GMiI1KWYoR3pWMlduVz1JJXRydWVHRiYtRjU2Iy9GMiI1PXhHelYyV25XPQ==KiQtSSdNYXRyaXhHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy9JJCVpZEdGKCI1NUhGelYyV25XPSEiIg==Ly0mSSdWZWN0b3JHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0knY29sdW1uR0YpNiMvSSQlaWRHRikiNTVURnpWMlduVz0tSTBkZWxheURvdFByb2R1Y3RHNiRGJy9JK21vZHVsZW5hbWVHRilJLFR5cGVzZXR0aW5nR0YmNiUqJC1JJ01hdHJpeEdGJjYjL0YuIjVJVUZ6VjJXblc9ISIiLUYkNiMvRi4iNV1WRnpWMlduVz1JJXRydWVHRic=f) oscillating force: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matrix form of solution:Ly1JMGRlbGF5RG90UHJvZHVjdEc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nRzYkRiZJKF9zeXNsaWJHRik2JS1JJ01hdHJpeEdGKzYjL0kkJWlkR0YpIjUtKmYjelYyV25XPS0mSSdWZWN0b3JHRis2I0knY29sdW1uR0YpNiMvRjIiNUErRXpWMlduVz1JJXRydWVHRiYtRjU2Iy9GMiI1VSxFelYyV25XPQ==KiQtSSdNYXRyaXhHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy9JJCVpZEdGKCI1aTNFelYyV25XPSEiIg==Ly0mSSdWZWN0b3JHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0knY29sdW1uR0YpNiMvSSQlaWRHRikiNVVERXpWMlduVz0tSTBkZWxheURvdFByb2R1Y3RHNiRGJy9JK21vZHVsZW5hbWVHRilJLFR5cGVzZXR0aW5nR0YmNiUqJC1JJ01hdHJpeEdGJjYjL0YuIjVpRUV6VjJXblc9ISIiLUYkNiMvRi4iNSN5aSN6VjJXblc9SSV0cnVlR0YnThis reduces to the correct result for frequency 8LUklc3Vic0clKnByb3RlY3RlZEc2JS9JI0YwRzYiIiRJIi9JJm9tZWdhR0YoIiIpSSVzb2w0R0YoApart from the overall factor of F0, the amplitude is:factor of 2 trips up the simplification of the powers!Students who did not first divide through by 2 ran into this 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 annoying.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JSFHAt 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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This ratio is pretty close to the value of the quality 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complex eigenvalue DE 3 tank mixing 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY+LUkjbWlHRiQ2JVErc29sZ2VuaGFuZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkobWZlbmNlZEdGJDYmLUYjNiotSSVtc3ViR0YkNiUtRiw2JVEieEYnRi9GMi1GIzYlLUkjbW5HRiQ2JFEiMUYnRjlGL0YyLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTlEsMC4zMzMzMzMzZW1GJy1GVTYlRlctRiM2JS1GZ242JFEiMkYnRjlGL0YyRmpuRl1vLUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkhGYW8vRk5GYm8tRlU2JUZXLUYjNiUtRmduNiRRIjNGJ0Y5Ri9GMkZqbi8lK2V4ZWN1dGFibGVHRj1GOUY5LyUlb3BlbkdRJyZsYW5nO0YnLyUmY2xvc2VHUScmcmFuZztGJ0ZccC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIRkpGTUZccC1GLDYlUSNjMUYnRi9GMkZccC1GLDYlUSdDb2x1bW5GJ0YvRjItRlA2JC1GIzYnLUZQNigtRiM2KC1JJ210YWJsZUdGJDY3LUkkbXRyR0YkNigtSSRtdGRHRiQ2KEZmbi8lKXJvd2FsaWduR1EhRicvJSxjb2x1bW5hbGlnbkdGW3MvJStncm91cGFsaWduR0Zbcy8lKHJvd3NwYW5HRmluLyUrY29sdW1uc3BhbkdGaW4tRmdyNigtRiM2Ky1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GXHQtSSZtZnJhY0dGJDYoRmZuRmlvLyUubGluZXRoaWNrbmVzc0dGaW4vJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGZXQvJSliZXZlbGxlZEdGPS1GNjYtUSgmbWludXM7RidGOUY7Rj5GQEZCRkRGRkZIRlt0Rl10LUYjNiotRiM2Ki1GX3Q2KEZmbi1GZ242JFEiNkYnRjlGYXRGY3RGZnRGaHQtRjY2LVExJkludmlzaWJsZVRpbWVzO0YnRjlGO0Y+RkBGQkZERkZGSEZhb0ZfcC1GIzYoLUZnbjYkUSJJRidGOS8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GJ0ZncC8lKXJlYWRvbmx5R0YxLyUwZm9udF9zdHlsZV9uYW1lR1EqMkR+T3V0cHV0RidGOUZedkZncEZhdkZjdkY5RmZ1LUkmbXNxcnRHRiQ2I0ZkcEZedkZncEZhdkZjdkY5Rl52RmdwRmF2RmN2RjlGaXJGXHNGXnNGYHNGYnMtRmdyNigtRiM2K0Zoc0ZedC1GNjYtUSIrRidGOUY7Rj5GQEZCRkRGRkZIRlt0Rl10Rl11Rl52RmdwRmF2RmN2RjlGaXJGXHNGXnNGYHNGYnNGaXJGXHNGXnMtRmRyNigtRmdyNigtRl90NihGZm5GZHBGYXRGY3RGZnRGaHRGaXJGXHNGXnNGYHNGYnNGaXZGZHNGaXJGXHNGXnMtRmRyNihGZnJGZnJGZnJGaXJGXHNGXnMvJSZhbGlnbkdRJWF4aXNGJy9GanJRKWJhc2VsaW5lRicvRl1zRmV0L0Zfc1EnfGZybGVmdHxockYnLyUvYWxpZ25tZW50c2NvcGVHRjEvJSxjb2x1bW53aWR0aEdRJWF1dG9GJy8lJndpZHRoR0ZkeC8lK3Jvd3NwYWNpbmdHUSYxLjBleEYnLyUuY29sdW1uc3BhY2luZ0dRJjAuOGVtRicvJSlyb3dsaW5lc0dRJW5vbmVGJy8lLGNvbHVtbmxpbmVzR0ZfeS8lJmZyYW1lR0ZfeS8lLWZyYW1lc3BhY2luZ0dRLDAuNGVtfjAuNWV4RicvJSplcXVhbHJvd3NHRj0vJS1lcXVhbGNvbHVtbnNHRj0vJS1kaXNwbGF5c3R5bGVHRj0vJSVzaWRlR1EmcmlnaHRGJy8lMG1pbmxhYmVsc3BhY2luZ0dGXHlGXnZGZ3BGYXZGY3ZGOUY5L0krbXNlbWFudGljc0dGJFEnTWF0cml4RicvRmpwUSJbRicvRl1xUSJdRidGYnpGXW9GZm5GZ3BGOUY5Rl13LUYsNiVRI2MyRidGL0YyRlxwLUYsNiVRI1gxRidGL0YyRl13LUYsNiVRI2MzRidGL0YyRlxwLUYsNiVRI1gyRidGL0YyLUY2Ni1RIjtGJ0Y5RjtGYG9GQEZCRkRGRkZIRmFvRk1GXHAtRiw2JVElZXFuc0YnRi9GMkY1LUYsNiVRJXN1YnNGJ0YvRjItRlA2JC1GIzYqLUYsNiVRInRGJ0YvRjJGX3EtRmduNiRGXG9GOUZdby1GLDYlUSRyaHNGJ0YvRjItRlA2JC1GIzYlLUYsNiVRIiVGJ0YvRjJGZ3BGOUY5RmdwRjlGOUZfcS1GUDYmLUYjNiktRmduNiRRIzE0RidGOUZdb0ZlXGxGXW9GZVxsRmdwRjlGOUZpcEZccUZncEY5LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkobWFjdGlvbkdGJDYkLUkobWZlbmNlZEdGJDYoLUYjNiUtSSNtaUdGJDYjUSFGJy1JJ210YWJsZUdGJDY3LUkkbXRyR0YkNigtSSRtdGRHRiQ2KC1GIzYjLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lKXJvd2FsaWduR0Y5LyUsY29sdW1uYWxpZ25HRjkvJStncm91cGFsaWduR0Y5LyUocm93c3BhbkdGSC8lK2NvbHVtbnNwYW5HRkgtRkE2KC1GIzYkLUkjbW9HRiQ2LVEqJnVtaW51czA7RidGSS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGam4vJSlzdHJldGNoeUdGam4vJSpzeW1tZXRyaWNHRmpuLyUobGFyZ2VvcEdGam4vJS5tb3ZhYmxlbGltaXRzR0Zqbi8lJ2FjY2VudEdGam4vJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0Zpby1JJm1mcmFjR0YkNihGRS1GIzYjLUZGNiRRIjJGJ0ZJLyUubGluZXRoaWNrbmVzc0dGSC8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZocC8lKWJldmVsbGVkR0ZqbkZMRk5GUEZSRlQtRkE2KC1GIzYkRlotRl1wNihGRS1GIzYlRmFwLUYjNiRGNi1JJm1zcXJ0R0YkNiMtRiM2Iy1GRjYkUSIzRidGSUY2RmRwRmZwRmlwRltxRkxGTkZQRlJGVEZMRk5GUC1GPjYoLUZBNigtRiM2Iy1GXXA2KEZFRmpxRmRwRmZwRmlwRltxRkxGTkZQRlJGVEZWLUZBNigtRiM2I0ZhcUZMRk5GUEZSRlRGTEZORlAtRj42KEZARkAtRkE2KC1GIzYjLUZGNiRRIjBGJ0ZJRkxGTkZQRlJGVEZMRk5GUC8lJmFsaWduR1ElYXhpc0YnL0ZNUSliYXNlbGluZUYnL0ZPRmhwL0ZRUSd8ZnJsZWZ0fGhyRicvJS9hbGlnbm1lbnRzY29wZUdRJXRydWVGJy8lLGNvbHVtbndpZHRoR1ElYXV0b0YnLyUmd2lkdGhHRmF0LyUrcm93c3BhY2luZ0dRJjEuMGV4RicvJS5jb2x1bW5zcGFjaW5nR1EmMC44ZW1GJy8lKXJvd2xpbmVzR1Elbm9uZUYnLyUsY29sdW1ubGluZXNHRlx1LyUmZnJhbWVHRlx1LyUtZnJhbWVzcGFjaW5nR1EsMC40ZW1+MC41ZXhGJy8lKmVxdWFscm93c0dGam4vJS1lcXVhbGNvbHVtbnNHRmpuLyUtZGlzcGxheXN0eWxlR0Zqbi8lJXNpZGVHUSZyaWdodEYnLyUwbWlubGFiZWxzcGFjaW5nR0ZpdEY2RkkvSSttc2VtYW50aWNzR0YkUSdNYXRyaXhGJy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Zfdi8lK2FjdGlvbnR5cGVHUS5ydGFibGVhZGRyZXNzRictRiM2JEZaRkUvJTFzdXBlcnNjcmlwdHNoaWZ0R0Zjc0Y2LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkobWZlbmNlZEdGJDYmLUYjNiktSSNtaUdGJDYlUSNjMUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JVEjYzJGJ0Y0RjdGOi1GMTYlUSNjM0YnRjRGNy8lK2V4ZWN1dGFibGVHRkJGPkY+LyUlb3BlbkdRJyZsYW5nO0YnLyUmY2xvc2VHUScmcmFuZztGJy1GOzYtUSJ+RidGPkZAL0ZERkJGRUZHRklGS0ZNRk8vRlNGUS1GOzYtUSI9RidGPkZARmBvRkVGR0ZJRktGTS9GUFEsMC4yNzc3Nzc4ZW1GJy9GU0Zmb0Zdby1JJW1zdXBHRiQ2JS1JKG1hY3Rpb25HRiQ2JC1GLDYoLUYjNictRjE2I1EhRictSSdtdGFibGVHRiQ2Ny1JJG10ckdGJDYoLUkkbXRkR0YkNigtRiM2Ji1JI21uR0YkNiRRIjFGJ0Y+RmVuLyUscGxhY2Vob2xkZXJHRjZGPi8lKXJvd2FsaWduR0ZkcC8lLGNvbHVtbmFsaWduR0ZkcC8lK2dyb3VwYWxpZ25HRmRwLyUocm93c3BhbkdGY3EvJStjb2x1bW5zcGFuR0ZjcS1GXHE2KC1GIzYnLUY7Ni1RKiZ1bWludXMwO0YnRj5GQEZgb0ZFRkdGSUZLRk0vRlBRLDAuMjIyMjIyMmVtRicvRlNGaHItSSZtZnJhY0dGJDYoRmBxLUYjNiUtRmFxNiRRIjJGJ0Y+RjRGNy8lLmxpbmV0aGlja25lc3NHRmNxLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmZzLyUpYmV2ZWxsZWRHRkJGZW5GZHFGPkZmcUZocUZqcUZcckZeci1GXHE2KC1GIzYnRmRyLUZbczYoRmBxLUYjNidGX3MtRiM2JUZicC1JJm1zcXJ0R0YkNiMtRiM2JC1GYXE2JFEiM0YnRj5GPkY+RmJwRjRGN0Zic0Zkc0Znc0Zpc0ZlbkZkcUY+RmZxRmhxRmpxRlxyRl5yRmZxRmhxRmpxLUZpcDYoLUZccTYoLUYjNiYtRltzNihGYHEtRiM2JUZqdEY0RjdGYnNGZHNGZ3NGaXNGZW5GZHFGPkZmcUZocUZqcUZcckZeckZgci1GXHE2KC1GIzYmRl90RmVuRmRxRj5GZnFGaHFGanFGXHJGXnJGZnFGaHFGanEtRmlwNihGW3FGW3EtRlxxNigtRiM2Ji1GYXE2JFEiMEYnRj5GZW5GZHFGPkZmcUZocUZqcUZcckZeckZmcUZocUZqcS8lJmFsaWduR1ElYXhpc0YnL0ZncVEpYmFzZWxpbmVGJy9GaXFGZnMvRltyUSd8ZnJsZWZ0fGhyRicvJS9hbGlnbm1lbnRzY29wZUdGNi8lLGNvbHVtbndpZHRoR1ElYXV0b0YnLyUmd2lkdGhHRmB3LyUrcm93c3BhY2luZ0dRJjEuMGV4RicvJS5jb2x1bW5zcGFjaW5nR1EmMC44ZW1GJy8lKXJvd2xpbmVzR1Elbm9uZUYnLyUsY29sdW1ubGluZXNHRlt4LyUmZnJhbWVHRlt4LyUtZnJhbWVzcGFjaW5nR1EsMC40ZW1+MC41ZXhGJy8lKmVxdWFscm93c0dGQi8lLWVxdWFsY29sdW1uc0dGQi8lLWRpc3BsYXlzdHlsZUdGQi8lJXNpZGVHUSZyaWdodEYnLyUwbWlubGFiZWxzcGFjaW5nR0Zod0ZicEZlbkY+Rj4vSSttc2VtYW50aWNzR0YkUSdNYXRyaXhGJy9GaG5RIltGJy9GW29RIl1GJ0ZeeS8lK2FjdGlvbnR5cGVHUS5ydGFibGVhZGRyZXNzRictRiM2JkZkckZgcUY0RjcvJTFzdXBlcnNjcmlwdHNoaWZ0R0ZjdkZdby1GLDYoLUYjNigtRmZwNjctRmlwNiYtRlxxNigtRiM2JC1GYXE2JFEjMTRGJ0Y+Rj5GZnFGaHFGanFGXHJGXnJGZnFGaHFGanEtRmlwNiYtRlxxNigtRiM2JEZhdkY+RmZxRmhxRmpxRlxyRl5yRmZxRmhxRmpxLUZpcDYmLUZccTYoRmF2RmZxRmhxRmpxRlxyRl5yRmZxRmhxRmpxRmR2Rmd2L0ZpcUZbeUZqdkZcd0Zed0Zhd0Zjd0Zmd0Zpd0ZceEZeeEZgeEZjeEZleEZneEZpeEZceS8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GJ0Zlbi8lKXJlYWRvbmx5R0Y2LyUwZm9udF9zdHlsZV9uYW1lR1EqMkR+T3V0cHV0RidGPkY+L0ZfeVEqQ29sVmVjdG9yRidGYXlGY3lGXlxsRmVuRj4=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkobWZlbmNlZEdGJDYmLUYjNiwtSSNtaUdGJDYlUSVzdWJzRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUZHNiQtRiM2NS1GTDYlUSNjMUYnRk9GUi1GLDYtUSI9RidGL0YyRjVGN0Y5RjtGPUY/L0ZCUSwwLjI3Nzc3NzhlbUYnL0ZFRmluLUkjbW5HRiQ2JFEiNkYnRi8tRiw2LVEiLEYnRi9GMi9GNkZRRjdGOUY7Rj1GP0ZBL0ZFUSwwLjMzMzMzMzNlbUYnLUZMNiVRI2MyRidGT0ZSRmVuLUYsNi1RKiZ1bWludXMwO0YnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yMjIyMjIyZW1GJy9GRUZccEZbb0Zfby1GTDYlUSNjM0YnRk9GUkZlbkZoby1GIzYqLUZcbzYkUSMxMEYnRi8tRiw2LVExJkludmlzaWJsZVRpbWVzO0YnRi9GMkY1RjdGOUY7Rj1GP0ZBRkQtSSZtc3FydEdGJDYjLUZcbzYkUSIzRidGLy8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GJy8lK2V4ZWN1dGFibGVHRjQvJSlyZWFkb25seUdGUS8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnRi9GX28tRkw2JVEkcmhzRidGT0ZSLUZHNiQtRiM2JS1GTDYlUStzb2xnZW5oYW5kRidGT0ZSRmJxRi9GLy1GRzYmLUYjNiUtRlxvNiRRIjFGJ0YvRmJxRi9GLy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZicUYvRi9GX29GSy1GRzYkLUYjNjRGWEZlbkZbb0Zfb0Zlb0ZlbkZob0Zbb0Zfb0ZecEZlbkZob0ZhcEZfb0ZpcS1GRzYkLUYjNiRGYHJGL0YvLUZHNiYtRiM2JC1GXG82JFEiMkYnRi9GL0YvRmpyRl1zRi9GL0Zfb0ZLLUZHNiQtRiM2NEZYRmVuRltvRl9vRmVvRmVuRmhvRltvRl9vRl5wRmVuRmhvRmFwRl9vRmlxRmRzLUZHNiYtRiM2JEZccUYvRi9GanJGXXNGL0YvRmJxRi9GLy9GW3NRJyZsYW5nO0YnL0Zec1EnJnJhbmc7RictRiw2LVEiO0YnRi9GMkZib0Y3RjlGO0Y9Rj9GQUZqbkYrLUZMNiVRK3NvbGl2cGhhbmRGJ0ZPRlItRiw2LVEqJmNvbG9uZXE7RidGL0YyRjVGN0Y5RjtGPUY/RmhuRmpuLUZMNiVRKXNpbXBsaWZ5RidGT0ZSLUZHNiQtRiM2JS1GTDYlUSIlRidGT0ZSRmJxRi9GL0ZicUYvJSFHFirst we find the general solution and the IVP solution and plot the 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnc29saXZwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkwtRiw2JVErc29saXZwaGFuZEYnRi9GMi8lK2V4ZWN1dGFibGVHRj1GOQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRJ3NvbGl2cEYnRi9GMi8lK2V4ZWN1dGFibGVHRj1GOQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFHplot 5: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. DE eigenbasis, 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adapting the coordinates to the eigenvectors (diagonalization of A)gridplot definitions (only need to use, not understand code)This modifies the command plot_plus_grid by Klaus Volpert from the gridlines worksheet:
http://www.math.villanova.edu/archives/maple/misc/gridline.mwsgrid_plot:=proc(a,b,c,d, m,n)
local i,j,v,h, p, ai, bi, ci,di,V,H;
#with(plots,display):
ai:=ceil(a): bi:=floor(b): ci:=ceil(c): di:=floor(d):
V:=[];
for i from ai to bi by m do
v[i]:= plot([i,t,t=c..d], thickness=0, color=black, linestyle=2):
V:=[op(V),v[i]];
od:
H:=[];
for j from ci to di by n do
h[j]:= plot([t,j,t=a..b], thickness=0, color=black, linestyle=2):
H:=[op(H),h[j]];
od:
plots[display]([op(V), op(H)]);
end;new_grid_plot:=proc(X,Y,a,b,c,d, m,n)
local i,j,v,h, p, ai, bi, ci,di,V,H;
#with(plots,display):
ai:=ceil(a): bi:=floor(b): ci:=ceil(c): di:=floor(d):
V:=[];
for i from ai to bi by m do
v[i]:= plot([i*X[1]+t*Y[1],i*X[2]+t*Y[2],t=c..d], thickness=0, color=blue, linestyle=1,thickness=1):
V:=[op(V),v[i]];
od:
H:=[];
for j from ci to di by n do
h[j]:= plot([t*X[1]+j*Y[1],t*X[2]+j*Y[2],t=a..b], thickness=0, color=blue, linestyle=1,thickness=1):
H:=[op(H),h[j]];
od:
plots[display]([op(V), op(H)]);
end;This helps visualize the coordinate system associated with a new basis of the plane: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display(
grid_plot(-10.5,3.5, -2.5,8.5,1,1), # EDIT window
new_grid_plot([1,2],[3,-1], # EDIT new basis vectors
-2.5,2.5,-3.5,3.5,1,1),scaling=constrained); gridplots:=%:
JSFHThe coefficient matrix: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 inverse matrix takes us from the old coordinates to the new coordinates.eigenvectors by handLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==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So 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 or LUkobWZlbmNlZEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXJvd0dGJDYmLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRjsvJSpzeW1tZXRyaWNHRjsvJShsYXJnZW9wR0Y7LyUubW92YWJsZWxpbWl0c0dGOy8lJ2FjY2VudEdGOy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYvNiRRIjJGJ0YyRjJGMi8lJW9wZW5HUScmbGFuZztGJy8lJmNsb3NlR1EnJnJhbmc7Ric= is an eigenvector basis for this eigenspace.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So 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 is an eigenvector basis for this eigenspace.JSFHJSFHNow we evaluate the new components of the initial position.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So the new coordinates are LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtbkdGJDYkUSIyRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiM0YnRjRGNEY0LyUlb3BlbkdRJyZsYW5nO0YnLyUmY2xvc2VHUScmcmFuZztGJ0Y0. We check this linear combination of the new basis vector.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The general solution is a linear combination of these eigenvectors with coefficients which are exponentials whose rate coefficients are the eigenvalues. The coefficients are the new coordinates of the initial position.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The two characteristic times are 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. The bigger one determines the longer decay.The first coordinate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ decays to the one percent level in 4.6 characteristic timesLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRJDQuNkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUYvNiRRIjJGJ0YyLyUnaXRhbGljR1EldHJ1ZUYnL0YzUSdpdGFsaWNGJy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRC8lKWJldmVsbGVkR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRklGMg==Now we do the DEplot but since I have to provide the answer key as a PDF, I will use Maple to annotate the plot and draw the parallelogram. You only need to do the basic plot and print it out and use a pencil or pen to complete it.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 for the direction field with the eigenvectors and new coordinate axes, and the parallelogram projection of the initial vector along those axes.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The arrows line up along the eigendirections as they should. Arrows point towards origin along those lines, indicating negative eigenvalues.e)
These are the actual solution curves versus time, needed for the final part.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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of 8 required plots1c,d. plots 1 and 2:
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pixels merge by this time,
amplitude \342\211\210 2.85 looks right
1f,i. plots 3 and 4:
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90 degrees out of phase (1/4 cycle),
blue lagging behind (later in time),
characteristic of near resonancegreen is the natural frequency
blue is the peak frequency
2e. plot 5; 3d plot 6
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for viewing windowyellow solution curve approaches origin quickly,
reaches the one percent level of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ at about time 2.3LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
3f. plots 7 and 8:
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4.6 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 max shown for 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
pledgeWhen you have completed the exam, please read and sign the dr bob integrity pledge and hand this test sheet stapled on top of your answer sheets as a cover page, with the first test page facing up:"During this examination, all work has been my own. I have read the long instructions on the class web page. I give my word that I have not resorted to any ethically questionable means of improving my grade or anyone else's on this examination and that I have not discussed this exam with anyone other than my instructor, nor will I until after the exam period is terminated for all participants."
Signature: Date: TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0Mzc5Mjg3ODM4WColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjLCYlI2MxRyIiKSUjYzJHIiNrLCZGJyEja0YpRihGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0Mzc5Mjg3NzE4WColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjIiRJIkYmRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0Mzc5MjczNTEwWCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMjIiIiIiQ/JiNGJyIjbCMhIiJGKkYmRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0Mzc5Mjc0MTEwWColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjJSNjMUclI2MyR0YlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0Mzc5MjY4MzM0WColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjIyEiKCIiJSMiIipGKEYlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0Mzc5MjYwMjYyWColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjLCYqJiwmKiQlJm9tZWdhRyIiIyEiIyIkTyIiIiJGLiUjYzFHRi5GLiomJSNjMkdGLkYqRi4iIiksJiomRi9GLkYqRi4hIikqJkYoRi5GMUYuRi5GJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0Mzc5MjYwMTQyWColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjJSNGMEdGJkYlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0Mzc5MjYxMzQyWCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMsJComLCYqJCUmb21lZ2FHIiIjIiIiISNvRixGLCwoKiRGKiIiJUYsKiRGKkYrISQ/IiIlQ1lGLCEiIiNGNEYrLCQqJkYqRixGLkY0RissJEY3ISIjRiZGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0Mzc5MjYyNTQyWColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjJSNjMUclI2MyR0YlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0Mzc5MjYzMzgyWColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjLCYqKCwmKiQlJm9tZWdhRyIiIyIiIiEjb0YsRiwsKCokRioiIiVGLCokRipGKyEkPyIiJUNZRiwhIiIlI0YwR0YsI0Y0RisqKEYqRixGLkY0RjVGLCEiIywmRjdGK0YnRjZGJQ==