MAT2705-02/06 09F 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. Print requested technology plots, annotate them appropriately by hand and attach to the relevant problems.
1. The displacement LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== of an underdamped harmonic oscillator system satisfies
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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSIuRidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZURj0=
Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk= , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjRGJ0Y5Rjk= , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIzQwRidGOUY5 and the initial conditions 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 .
a) What are the natural frequency LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JJW1yb3dHRiQ2JC1JI21uR0YkNiRRIjBGJ0YyRjIvJS9zdWJzY3JpcHRzaGlmdEdGOw== , natural decay time LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXJvd0dGJDYkLUkjbWlHRiQ2JVEmJnRhdTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjUtRiw2JC1JI21uR0YkNiRRIjBGJ0Y1RjUvJS9zdWJzY3JpcHRzaGlmdEdGPQ== , the quality factor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GIzYoLUYsNiNRIUYnLUklbXN1YkdGJDYlLUYsNiVRJyYjOTY5O0YnL0YwRj1GOS1GIzYkLUkjbW5HRiQ2JFEiMEYnRjlGOS8lL3N1YnNjcmlwdHNoaWZ0R0ZqbkZRLUYjNiZGUS1GVTYlLUYjNiQtRiw2JVEmJnRhdTtGJ0ZaRjlGOS1GIzYlRmduLUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GXHBGOUZbb0ZRRjlGUUY5RlFGOQ==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 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/LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJm9tZWdhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSVtcm93R0YkNiQtSSNtbkdGJDYkUSIxRidGMkYyLyUvc3Vic2NyaXB0c2hpZnRHUSIwRic= for this decaying sinusoidal solution? (Give both exact and numeric values to 3 decimal places.)c) 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.
Find the initial value problem solution by hand and evaluate the asymptotic value 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.
Make a plot in an appropriate viewing window (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RLyZHcmVhdGVyRXF1YWw7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSIwRidGOUY5 up to 5 decay times) showing both the solution and its horizontal asymptote together, showing clearly the initial oscillating behavior and the approach to the asymptotic value. Does the period of the damped oscillation in your plot roughly agree with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ ? Support your claim.d) 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.
Find the initial value problem solution by hand (but check with Maple!)
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) and express the phase shift in radians, degrees and cycles. Make a single plot in an appropriate viewing window 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 both the steady state solution and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSRzaW5GJy9GMEY9RjktSShtZmVuY2VkR0YkNiQtRiM2Ji1JI21uR0YkNiRRIjZGJ0Y5RjUtRiw2JVEidEYnRi9GMkY5RjlGOQ== (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 the phase shift you calculated in fractions of a cycle, given that the driving sine is itself shifted 1/4 cycle to the right (later in time) from the vertical axis at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=? Explain.e) 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.Explore resonance for this system by finding the steady state solution by hand, where the nonegative frequency LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy of the driving force function is a parameter. f) Evaluate the steady state amplitude function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRJyYjOTY5O0YnL0YwUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0Y/Rj8tSSNtb0dGJDYtUSJ+RidGPy8lJmZlbmNlR0Y+LyUqc2VwYXJhdG9yR0Y+LyUpc3RyZXRjaHlHRj4vJSpzeW1tZXRyaWNHRj4vJShsYXJnZW9wR0Y+LyUubW92YWJsZWxpbWl0c0dGPi8lJ2FjY2VudEdGPi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlVGPw==and use calculus to find the exact and numerical value of the frequency LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiM2JC1GLDYlUSgmb21lZ2E7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y6LUYjNiQtRiw2JVEicEYnL0Y4USV0cnVlRicvRjtRJ2l0YWxpY0YnRjovJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0YrRjo= and the amplitude LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUklbXN1YkdGJDYlLUYsNiVRKCZvbWVnYTtGJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRictRiM2JC1GLDYlUSJwRidGL0YyRkIvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZCRkJGQg== where it has its peak value for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RLyZHcmVhdGVyRXF1YWw7RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkktSSNtbkdGJDYkUSIwRidGMkYy.
What is the numerical value of the ratio 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? How does it compare to the quality factor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ? Does LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiNkYnL0YzUSdub3JtYWxGJ0Y+Rj5GPg== agree with A from part d)?g) Plot this amplitude function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRKCZvbWVnYTtGJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRidGP0Y/Rj8= in an appropriate window (showing the behavior of the entire function for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RLyZHcmVhdGVyRXF1YWw7RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkktSSNtbkdGJDYkUSIwRidGMkYy) together with the constant functions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiNRIUYnLUYjNiZGOi1JJW1zdWJHRiQ2JS1GLDYlUScmIzk2OTtGJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRictRiM2JC1JI21uR0YkNiRRIjBGJ0ZHRkcvJS9zdWJzY3JpcHRzaGlmdEdGTkY6RkdGOkZHRkdGOkZH, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiNRIUYnLUYjNiZGOi1JJW1zdWJHRiQ2JS1GLDYlUScmIzk2OTtGJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRictRiM2JC1GLDYlUSJwRidGL0YyRkcvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y6RkdGOkZHRkdGOkZH and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMEYnL0YzUSdub3JtYWxGJ0Y+Rj4tRiw2I1EhRidGPg== and hand annotate on your axes the values of these frequencies and amplitudes and indicate the points on the curve where these values are obtained.
2. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2I1EhRictSShtYWN0aW9uR0YkNiQtSShtZmVuY2VkR0YkNigtRiM2JkYrLUknbXRhYmxlR0YkNjYtSSRtdHJHRiQ2Ji1JJG10ZEdGJDYoLUYjNigtSSVtc3ViR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0ZMUSdub3JtYWxGJ0ZULyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSInRidGVC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGaW4vJSlzdHJldGNoeUdGaW4vJSpzeW1tZXRyaWNHRmluLyUobGFyZ2VvcEdGaW4vJS5tb3ZhYmxlbGltaXRzR0Zpbi8lJ2FjY2VudEdGaW4vJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR1EmMC4wZW1GJy1GWjYtUSIoRidGVC9GaG5GSkZqbi9GXW9GSkZeb0Zgb0Zib0Zkby9GZ29RLDAuMTY2NjY2N2VtRicvRmpvRmJwLUYsNiVRInRGJ0ZIRkstRlo2LVEiKUYnRlRGX3BGam5GYHBGXm9GYG9GYm9GZG9GYXBGY3BGVC8lKXJvd2FsaWduR0YuLyUsY29sdW1uYWxpZ25HRi4vJStncm91cGFsaWduR0YuLyUocm93c3BhbkdGUy8lK2NvbHVtbnNwYW5HRlNGanBGXHFGXnEtRjs2Ji1GPjYoLUYjNigtRkM2JUZFLUYjNiQtRlE2JFEiMkYnRlRGVEZWRllGXHBGZHBGZ3BGVEZqcEZccUZecUZgcUZicUZqcEZccUZecS8lJmFsaWduR1ElYXhpc0YnL0ZbcVEpYmFzZWxpbmVGJy9GXXFRJ2NlbnRlckYnL0ZfcVEnfGZybGVmdHxockYnLyUvYWxpZ25tZW50c2NvcGVHRkovJSxjb2x1bW53aWR0aEdRJWF1dG9GJy8lJndpZHRoR0Zecy8lK3Jvd3NwYWNpbmdHUSYxLjBleEYnLyUuY29sdW1uc3BhY2luZ0dRJjAuOGVtRicvJSlyb3dsaW5lc0dRJW5vbmVGJy8lLGNvbHVtbmxpbmVzR0Zpcy8lJmZyYW1lR0Zpcy8lLWZyYW1lc3BhY2luZ0dRLDAuNGVtfjAuNWV4RicvJSplcXVhbHJvd3NHRmluLyUtZXF1YWxjb2x1bW5zR0Zpbi8lLWRpc3BsYXlzdHlsZUdGaW4vJSVzaWRlR1EmcmlnaHRGJy8lMG1pbmxhYmVsc3BhY2luZ0dGZnNGK0ZURlQvSSttc2VtYW50aWNzR0YkUSpDb2xWZWN0b3JGJy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZcdS8lK2FjdGlvbnR5cGVHUS5ydGFibGVhZGRyZXNzRictRlo2LVEiPUYnRlRGZ25Gam5GXG9GXm9GYG9GYm9GZG8vRmdvUSwwLjI3Nzc3NzhlbUYnL0Zqb0Zcdi1GMDYkLUYzNigtRiM2JkYrLUY4NjYtRjs2Jy1GPjYoLUYjNiUtRlo2LVEqJnVtaW51czA7RidGVEZnbkZqbkZcb0Zeb0Zgb0Zib0Zkby9GZ29RLDAuMjIyMjIyMmVtRicvRmpvRmB3LUZRNiRRIjVGJ0ZURlRGanBGXHFGXnFGYHFGYnEtRj42KC1GIzYkLUZRNiRRIjNGJ0ZURlRGanBGXHFGXnFGYHFGYnFGanBGXHFGXnEtRjs2Jy1GPjYoLUYjNiVGXHctRlE2JFEiNkYnRlRGVEZqcEZccUZecUZgcUZicS1GPjYoRk5GanBGXHFGXnFGYHFGYnFGanBGXHFGXnFGYXJGZHJGZnJGaHJGanJGXHNGX3NGYXNGZHNGZ3NGanNGXHRGXnRGYXRGY3RGZXRGZ3RGanRGK0ZURlQvRl11USdNYXRyaXhGJ0ZfdUZidUZneEZldS1GWjYtUSJ+RidGVEZnbkZqbkZcb0Zeb0Zgb0Zib0Zkby9GZ29GW3BGaW8tRjA2JC1GIzYoRistRiM2KEYrLUYjNiZGKy1GIzYlLUYzNigtRiM2JkYrLUY4NjYtRjs2Ji1GPjYoLUYjNidGQkZccEZkcEZncEZURmpwRlxxRl5xRmBxRmJxRmpwRlxxRl5xLUY7NiYtRj42KC1GIzYnRmpxRlxwRmRwRmdwRlRGanBGXHFGXnFGYHFGYnFGanBGXHFGXnFGYXJGZHJGZnJGaHJGanJGXHNGX3NGYXNGZHNGZ3NGanNGXHRGXnRGYXRGY3RGZXRGZ3RGanRGK0ZURlRGXHVGX3VGYnVGXHUtRlo2LVEiLEYnRlRGZ24vRltvRkpGXG9GXm9GYG9GYm9GZG9GXHkvRmpvUSwwLjMzMzMzMzNlbUYnRlRGaXhGVEYrLUYwNiQtRjM2KC1GIzYmRistRjg2Ni1GOzYmLUY+NigtRiM2J0ZCRlxwLUZRNiRGWEZURmdwRlRGanBGXHFGXnFGYHFGYnFGanBGXHFGXnEtRjs2Ji1GPjYoLUYjNidGanFGXHBGXVxsRmdwRlRGanBGXHFGXnFGYHFGYnFGanBGXHFGXnFGYXJGZHJGZnJGaHJGanJGXHNGX3NGYXNGZHNGZ3NGanNGXHRGXnRGYXRGY3RGZXRGZ3RGanRGK0ZURlRGXHVGX3VGYnVGXHVGZXVGaHVGVEYrLUYwNiQtRjM2KC1GIzYmRistRjg2Ni1GOzYmRmV4RmpwRlxxRl5xLUY7NiYtRj42KEZcckZqcEZccUZecUZgcUZicUZqcEZccUZecUZhckZkckZmckZockZqckZcc0Zfc0Zhc0Zkc0Znc0Zqc0ZcdEZedEZhdEZjdEZldEZndEZqdEYrRlRGVEZcdUZfdUZidUZcdUZldUYrRlRGZXVGK0ZULUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=a) Write this system of differential equations for the vector variable 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 = <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> AND its initial conditions in scalar form (use arrow notation for vectors!).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) Express sinusoidal factors in each of the two unknowns as phase-shifted cosines in terms of their amplitudes and phase shifts. Determine the upper and lower envelope curves for each such function from this step (plus or minus amplitude times exponential factor from solutions).e) Make a single plot showing the original two solution curves (this is a check of your amplitude) and their envelope curves versus LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= on the same axes for at least 5 characteristic times of the decaying exponential factor in these expressions (i.e., in an appropriate viewing window). Hand annotate each solution curve with its variable name.f) It appears that the first intersection point of the graphs of the two variables in this plot coincides with the point where one of the two variables touches its envelope curve for the first time. Find this common intersection point exactly and numerically to several digits accuracy and show that indeed the two functions and the relevant envelope curve all have the same exact value at that point.
Does your calculated value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= where this occurs seem consistent with your plot? Explain3. 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.a) Identify the coefficient matrix and find a new basis for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0Y+LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y+ consisting of eigenvectors 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 of this matrix using the standard hand recipe. Order the real eigenvalues 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 by decreasing value.b) Evaluate the new coordinates LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiQtSSNtbkdGJDYkUSIxRicvRjtRJ25vcm1hbEYnRkMvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIixGJ0ZDLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y5LyUpc3RyZXRjaHlHRk4vJSpzeW1tZXRyaWNHRk4vJShsYXJnZW9wR0ZOLyUubW92YWJsZWxpbWl0c0dGTi8lJ2FjY2VudEdGTi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYxNiVGMy1GIzYkLUZANiRRIjJGJ0ZDRkNGRUZDRkMvJSVvcGVuR1EnJmxhbmc7RicvJSZjbG9zZUdRJyZyYW5nO0YnRkM= of the point 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 with respect to this basis of eigenvectors.c) Use technology to plot 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.d) Plot the two variables versus LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=for an appropriate viewing window for this initial value problem based on the longest characteristic time. Explain your window choice.The first variable LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ has a local maximum obvious in your plot. Find its coordinates and the corresponding value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= exactly and approximately. 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 me as though it were material for a job interview (you're fired! or you're hired! ?). In a real world technical job, you need to be able to write coherent technical reports that other people can follow.solution1. damped harmonic oscillatorJSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUShyZXN0YXJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiO0YnRj0vJSZmZW5jZUdGPC8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y8LyUqc3ltbWV0cmljR0Y8LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjwvJSdhY2NlbnRHRjwvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJ0Y6Rj0=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)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) no 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c) constant 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 right, each period is about 1 time unit long for a total of about two and a half oscillations before it disappears.d)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 the merging of pixels after about 4 or 5 times the characteristic time 1/2.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The coefficients are 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 so: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 to check they do the same thing: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 we plot the solution (red) together with the steady state solution (blue) and a rescaled driving function (green) to compare the phase shift.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Let's recall the natural frequency and quality 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oscillating 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ratio is pretty close to the value of the quality factor.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. complex eigenvalue DEJSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUShyZXN0YXJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiO0YnRj0vJSZmZW5jZUdGPC8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y8LyUqc3ltbWV0cmljR0Y8LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjwvJSdhY2NlbnRHRjwvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJ0Y6Rj0=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First we find the general solution and the IVP solution and plot the 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=eigenvector solution 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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 gives exactly these eigenvectors, though it randomly reorders them. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1GLDYtUSIsRidGL0YyL0Y2RkxGN0Y5RjtGPUY/RkEvRkVRLDAuMzMzMzMzM2VtRictRkc2JVEjQkJGJ0ZKRk0tRiw2LVEqJmNvbG9uZXE7RidGL0YyRjVGN0Y5RjtGPUY/L0ZCUSwwLjI3Nzc3NzhlbUYnL0ZFRmZuLUZHNiVRLUVpZ2VudmVjdG9yc0YnRkpGTS1JKG1mZW5jZWRHRiQ2JC1GIzYlLUZHNiVRIkFGJ0ZKRk0vJStleGVjdXRhYmxlR0Y0Ri9GLy1GRzYjUSFGJ0Zjb0YvLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY1LUkjbWlHRiQ2JVEtc29sZ2VuYnloYW5kRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSShtZmVuY2VkR0YkNiYtRiM2Jy1GLDYlUSN4MUYnRi9GMi1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNiVRI3gyRidGL0YyLyUrZXhlY3V0YWJsZUdGPUY5RjkvJSVvcGVuR1EnJmxhbmc7RicvJSZjbG9zZUdRJyZyYW5nO0YnLUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkhGZW4vRk5GZm4tRjY2LVEiPUYnRjlGO0Y+RkBGQkZERkZGSEZKRk1GZG8tRiw2JVEjYzFGJ0YvRjJGZG8tSSVtc3VwR0YkNiUtRjY2LVEvJkV4cG9uZW50aWFsRTtGJ0Y5L0Y8USZ1bnNldEYnL0Y/RmVwL0ZBRmVwL0ZDRmVwL0ZFRmVwL0ZHRmVwL0ZJRmVwRmVuL0ZOUSwwLjExMTExMTFlbUYnLUYjNiotRlA2JC1GIzYtLUY2Ni1RKiZ1bWludXMwO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZocS1JI21uR0YkNiRRIjJGJ0Y5LUY2Ni1RIitGJ0Y5RjtGPkZARkJGREZGRkhGZ3FGaXEtRltyNiRRIjNGJ0Y5RmRvLUYsNiVRIklGJ0YvRjJGLy8lK2ZvcmVncm91bmRHUStbMCwxNjAsODBdRicvJSxwbGFjZWhvbGRlckdGMS8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdGMUYyRjlGZG8tRiw2JVEidEYnRi9GMkYvRmdyRmpyRlxzRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRmRvLUYsNiVRI2IyRidGL0YyRl5yLUYsNiVRI2MyRidGL0YyRmRvLUZfcDYlRmFwLUYjNiYtRlA2JC1GIzYpRmRxRmpxLUY2Ni1RKCZtaW51cztGJ0Y5RjtGPkZARkJGREZGRkhGZ3FGaXFGYXJGZG9GZHJGOUY5RmRvRl5zRjlGYXNGZG8tRiw2JVEjYjNGJ0YvRjJGXG9GOQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEoSVZQZXFuc0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRJXN1YnNGJ0YvRjItSShtZmVuY2VkR0YkNiQtRiM2MS1GLDYlUSJ0RidGL0YyLUY2Ni1RIj1GJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUkjbW5HRiQ2JFEiMEYnRjktRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTlEsMC4zMzMzMzMzZW1GJy1GLDYlUSN4MUYnRi9GMkZaLUZobjYkUSIxRidGOUZbby1GLDYlUSN4MkYnRi9GMkZaLUZobjYkUSIyRidGOUZbby1GLDYlUS1zb2xnZW5ieWhhbmRGJ0YvRjIvJStleGVjdXRhYmxlR0Y9RjlGOUZicEY5LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkjbWlHRiQ2JVEoY29udmVydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYoLUYsNiVRJHJoc0YnRi9GMi1GNjYkLUYjNiQtRiw2JVEoSVZQZXFuc0YnRi9GMi9GM1Enbm9ybWFsRidGRC1JI21vR0YkNi1RIixGJ0ZELyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkwvJSpzeW1tZXRyaWNHRkwvJShsYXJnZW9wR0ZMLyUubW92YWJsZWxpbWl0c0dGTC8lJ2FjY2VudEdGTC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRJWxpc3RGJ0YvRjIvJStleGVjdXRhYmxlR0ZMRkRGRC1GRzYtUSI7RidGREZKRk1GT0ZRRlNGVUZXRlkvRmduUSwwLjI3Nzc3NzhlbUYnLUYsNiVRL0dlbmVyYXRlTWF0cml4RidGL0YyLUY2NiQtRiM2Jy1GLDYlUSIlRidGL0YyRkYtRjY2Ji1GIzYmLUYsNiVRI2MxRidGL0YyRkYtRiw2JVEjYzJGJ0YvRjJGREZELyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRlxvRkRGREZeby1GRzYtUSJ+RidGREZKL0ZORkxGT0ZRRlNGVUZXRlkvRmduRmVuLUYsNiVRJEJCQkYnRi9GMi1GRzYtUSomY29sb25lcTtGJ0ZERkpGYHFGT0ZRRlNGVUZXL0ZaRmJvRmFvLUYsNiVRLURlbGV0ZUNvbHVtbkYnRi9GMi1GNjYkLUYjNidGam9GRi1JI21uR0YkNiRRIjNGJ0ZERlxvRkRGREZcb0ZELUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkobWZlbmNlZEdGJDYmLUYjNictSSNtaUdGJDYlUSNjMUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JVEjYzJGJ0Y0RjcvJStleGVjdXRhYmxlR0ZCRj5GPi8lJW9wZW5HUScmbGFuZztGJy8lJmNsb3NlR1EnJnJhbmc7RictRjs2LVEifkYnRj5GQC9GREZCRkVGR0ZJRktGTUZPL0ZTRlEtRjs2LVEiPUYnRj5GQEZdb0ZFRkdGSUZLRk0vRlBRLDAuMjc3Nzc3OGVtRicvRlNGY28tSSVtc3VwR0YkNiUtRjE2JVEkQkJCRidGNEY3LUYjNiYtRjs2LVEqJnVtaW51czA7RidGPkZARl1vRkVGR0ZJRktGTS9GUFEsMC4yMjIyMjIyZW1GJy9GU0ZhcC1JI21uR0YkNiRRIjFGJ0Y+RlhGPi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRjs2LUZcb0Y+RkBGXW9GRUZHRklGS0ZNRmJvRmRvLUY7Ni1RIi5GJ0Y+RkBGXW9GRUZHRklGS0ZNRk9GXm9GanAtRiw2Ji1GIzYnRmNwRjotRmRwNiRRIjJGJ0Y+RlhGPkY+RlpGZ25GWEY+LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYyLUkjbWlHRiQ2JVEtc29saXZwYnloYW5kRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRiw2JVElc3Vic0YnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYuLUYsNiVRI2MxRidGL0YyLUY2Ni1RIj1GJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUkjbW5HRiQ2JFEiMUYnRjktRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTlEsMC4zMzMzMzMzZW1GJy1GLDYlUSNjMkYnRi9GMkZaRmduLUYsNiNRIUYnRltvLUYsNiVRLXNvbGdlbmJ5aGFuZEYnRi9GMi8lK2V4ZWN1dGFibGVHRj1GOUY5LUY2Ni1RIjtGJ0Y5RjtGXm9GQEZCRkRGRkZIRl9vRk0tRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSEZfby9GTkZgby1GLDYlUSRyaHNGJ0YvRjItRlM2JC1GIzYlLUYsNiVRIiVGJ0YvRjJGXHBGOUY5LUY2Ni1RIjpGJ0Y5RjtGPkZARkJGREZGRkhGSkZNRmFwLUYsNiVRI1hYRidGL0YyRjUtRiw2JVEkbWFwRidGL0YyLUZTNiQtRiM2Jy1GLDYlUSZldmFsY0YnRi9GMkZbb0ZccUZccEY5RjlGXHBGOQ==This agrees with the previous expressions:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEpc2ltcGxpZnlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYlUSJYRidGL0YyLUkjbW9HRiQ2LVEoJm1pbnVzO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZULUYsNiVRI1hYRidGL0YyLyUrZXhlY3V0YWJsZUdGRUZBRkFGWkZB3. DE eigenbasis, 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We set up the problem by picking the eigenectors and eigenvalues to construct the matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.Qyk+SSNiMUc2Ii1JJDwsPkdJKF9zeXNsaWJHRiU2JCMiIiIiIiNGK0YrPkkjYjJHRiUtRic2JCIiJEYrRis+SSJCR0YlLUkkPHxncj5HRig2JEYkRi5GKyokRjMhIiI=Now we pick the eigenvalues and work backwards to the original matrix to have a simple problemPkkiQUc2Ii1JMGRlbGF5RG90UHJvZHVjdEc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiRJLFR5cGVzZXR0aW5nRzYkRihJKF9zeXNsaWJHRiQ2JC1GJjYkSSJCR0YkLUknTWF0cml4R0YsNiMvSSQlaWRHRiQiKl9VWl0iKiRGMSEiIg==The inverse matrix takes us from the old coordinates to the new coordinates.LUkwZGVsYXlEb3RQcm9kdWN0RzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJUkoX3N5c2xpYkdGKDYlKiRJIkJHRighIiItJkknVmVjdG9yR0YqNiNJJ2NvbHVtbkdGKDYjL0kkJWlkR0YoIipPdWJ1IkkldHJ1ZUdGJQ==So the new coordinates are 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. We check this linear combination of the new basis vectors:LCYtSSdDb2x1bW5HNiI2JEkiQkdGJSIiIiIiJS1GJDYkRiciIiMhIiI=The general solution is a linear combination of these eigenvectors with coefficients which are exponentials whose rate coefficients are the eigenvalues, although this was not requested.Ly1JJDwsPkdJKF9zeXNsaWJHNiI2JCZJInhHRiY2IyIiIiZGKTYjIiIjLCYtSTBkZWxheURvdFByb2R1Y3RHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YmSSxUeXBlc2V0dGluZ0c2JEYzRiU2JSomSSNDMUdGJkYrLUkkZXhwR0YmNiMsJEkidEdGJiEiJkYrSSNiMUdGJkkldHJ1ZUdGM0YrLUYxNiUqJkkjQzJHRiZGKy1GPDYjLCRGPyEjNUYrSSNiMkdGJkZCRis=Now we do theDEplot 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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The arrows line up along the eigendirections as they should. Arrows point towards origin along those lines, indicating negative eigenvalues.d)
These are the actual solution curves versus time, needed for the final part.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JSFHpledgeWhen 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: TTdSMApJNlJUQUJMRV9TQVZFLzE1MDQ3OTQzNlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiMiIyMiIiIiIiNGKEYmTTdSMApJNlJUQUJMRV9TQVZFLzE1MDQ3OTUwMFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiMiIyIiJCIiIkYmTTdSMApJNlJUQUJMRV9TQVZFLzE1MDQ3OTU2NFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjIyIiIiIiI0YoIiIkRihGJg==TTdSMApJNlJUQUJMRV9TQVZFLzE1MDQ3OTg4NFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjIyEiIyIiJiMiIiNGKSMiIidGKSMhIiJGKUYmTTdSMApJNlJUQUJMRV9TQVZFLzE1MDQ4MDg0NFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjISM2ISIjIiIkISIlRiY=TTdSMApJNlJUQUJMRV9TQVZFLzE3NDU1ODA3NlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiMiIyIiJSEiIkYmTTdSMApJNlJUQUJMRV9TQVZFLzE3NDU1ODY1MlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiMiIyEiIiIiJEYmTTdSMApJNlJUQUJMRV9TQVZFLzE3NjY4MTkwOFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiMiIyYlInhHNiMiIiImRig2IyIiI0YmTTdSMApJNlJUQUJMRV9TQVZFLzE3NDU1ODkwOFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiMiIywmKiYlI0MxRyIiIi0lJGV4cEc2IywkJSJ0RyEiJkYqI0YqIiIjKiYlI0MyR0YqLUYsNiMsJEYvISM1RioiIiQsJkYoRipGM0YqRiY=