Useful solutions to memorizetwo DE solutions to memorize, we will derive the solutions later, but use them throughout the course.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Exponentials describe growth and decay problems, while sines and cosines describe oscillatory behavior. In this course we will be interested in oscillations in time: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which represent phase-shifted cosines of amplitude LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== and phase-shift LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHRjFGMg== which we restrict to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkobWZlbmNlZEdGJDYmLUYjNiUtSSNtaUdGJDYlUScmIzk0ODtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUrZXhlY3V0YWJsZUdGNkY3RjcvJSVvcGVuR1EifGdyRicvJSZjbG9zZUdGPi1JI21vR0YkNi1RJiZsZXE7RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlUtRjE2JVElJnBpO0YnRjRGN0Y6Rjc=, namely less than 180\302\260 to the right or to the left along the horizonal axis. The argument of the cosine or sine function 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 or LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSS1GLDYlUSJ0RicvRjBRJXRydWVGJy9GM1EnaXRhbGljRidGMg== is called the phase of the oscillation, hence the phase-shifting by subtracting LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHRjFGMg==.Shifting the cosine graph left and rightWe will need to describe translated cosine graphs using an angle LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2KFEnJiM5NDg7RicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjQvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjlGNS9GOFEnbm9ybWFsRic= (called the phase shift) less than or equal to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2KFEnJiM5NjA7RicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjQvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjlGL0Y1RjdGOg== (180 degrees!) in absolute value:
The reason is so we can describe most efficiently horizontal translations of the cosine graph to characterize where the peaks occur in cosine/sine wave functions.
upper half circle: 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 positive acute angles [1st quadrant on unit circle]
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 positive obtuse angles [2nd quadrant on unit circle]lower half circle: 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 negative acute angles [4th quadrant on unit circle]
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 negative obtuse angles [3rd quadrant on unit 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example: finding angles in 4 quadrants for a given slope absolute 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gives an angle in the interval 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 for the direction of the point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYlLUYjNictSSNtaUdGJDYmUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNy9GO1Enbm9ybWFsRicvJSZmZW5jZUdGOS8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYmUSJ5RidGNEY3RjpGN0ZBRjdGQUY3RkE= from the origin.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 the standard cosine graph with its horizontal translations. First the standard cosine: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 add a sliding phase-shifted cosine: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
SSZkZWx0YUc2Ig==#
# The following code is autogenerated by the Explore command. Do not edit.
#
Explore:-Runtime:-Update( plot([cos(x), cos(x-delta)],x = -3/2*Pi .. 3/2*Pi,gridlines = true,color = [gray, red]), ':-Plot0', [':-delta'], ':-sliders'=["Sliderdelta"], ':-labels'=["Labeldelta"],':-isplot'=true, ':-handlers'=[delta=handlertab[delta]]);
#
# End of autogenerated code.
#
If you slide left or right more than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEnJiM5NjA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRJDE4MEYnRjItRjY2LVEmJmRlZztGJ0YyRjlGO0Y9Rj9GQUZDRkUvRkhRJjAuMGVtRicvRktGVC8lK2V4ZWN1dGFibGVHRjFGMg==, you repeat the same sequence of curves. At LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStleGVjdXRhYmxlR0YxRjI= or LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkMtSSNtaUdGJDYlUSUmcGk7RicvJSdpdGFsaWNHRjRGLy8lK2V4ZWN1dGFibGVHRjRGLw== the shifted curve is 180\302\260 out of phase. For positive phase shifts LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJiZsZXE7RidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHRjhGLy1GMzYtUSI8RidGL0Y2RjlGO0Y9Rj9GQUZDRkVGSC1GSzYlUSUmcGk7RidGTkYvLyUrZXhlY3V0YWJsZUdGOEYv, the peaks are to the right of the cosine peak at zero (later in time if the horizontal axis represents a time variable phase LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSS1GLDYlUSJ0RicvRjBRJXRydWVGJy9GM1EnaXRhbGljRicvJStleGVjdXRhYmxlR0YxRjI=). For negative phase shifts 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, the peaks are to the left of the cosine peak at zero (earlier in time if the horizontal axis represents a time variable phase LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSS1GLDYlUSJ0RicvRjBRJXRydWVGJy9GM1EnaXRhbGljRicvJStleGVjdXRhYmxlR0YxRjI=). At zero phase shift, the two cosines are "in phase".plot for above 4 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When the phase shift is a negative acute or obtuse angle, the peak is shifted left.
When the phase shift is a positive acute or obtuse angle, the peak is shifted right.animation of shifting of cosineThere are 26 frames in this video below the text. If you click on the plot and slowly move the video frame scroll bar through the frames to frame 13, you arrive at zero phase shift where the two cosine waves overlap. As you move left or right, the peak of the red cosine moves away from the vertical axis towards the edges of the plot. When it moves right and passes \317\200 for example, it moves out of the right edge and into the left edge, moving again to the right as it returns back to the origin as \316\264 increases from \317\200 to 2\317\200, during which it duplicates the range of the phase shift from LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LlEqJnVtaW51czA7RicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZFLUkjbWlHRiQ2JlEnJiM5NjA7RicvJSdpdGFsaWNHRjFGL0YyRi9GMg== to 0. It only makes sense to say the peak is ahead or behind by less then half a cycle (one period = 2\317\200) compared to the standard cosine wave.
When the phase shift is a negative acute or obtuse angle, the peak is shifted left.
When the phase shift is a positive acute or obtuse angle, the peak is shifted right.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRJnBsb3RzRidGL0YyL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIjpGJ0Y9LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlQ=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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JUYrLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRitGMUY0Solving systems of linear differential equationsBy the end of the course you will understand the mathematics of coupled systems of linear ordinary differential equations which describe coupled damped harmonic oscillator systems, for example, which could be a stick figure model of some structure or coupled simple electrical circuits, etc.A nice example is the study of horizontal vibrations of multistory buildings during an earthquake, although we will stop short of exploring this case. However, suppose we consider a simple system of two coupled DEs to see how Maple solves it. To have LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn for time be the independent variable instead of the default single variable calculus variable LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn it is enough to use function notation: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