73 lines
2.6 KiB
Markdown
73 lines
2.6 KiB
Markdown
---
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title: Homework 1
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allDay: true
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date: 2024-09-18
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endDate: null
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completed: 2024-09-26T09:55:06.474-04:00
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type: single
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startTime: 10:00
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endTime: 12:00
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---
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# Instructions
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#Homework
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Please do a written solution for problems 1 and 2. We will review them on Monday, Sept 16 in class prior to the assignment being due.
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Please upload a Jupyter Notebook for problems 3 and 4.
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Problems 1 and 2 are worth 10 points each, problems 3 and 4 are worth 15 points each.
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# Written Problems
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## Problem 1
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Please find the general solution of
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$$
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\bf{\dot{X}} =
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\begin{bmatrix}
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-1 & 5 & 2\\
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4 & -1 & -2\\
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0 & 0 & 6
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\end{bmatrix}
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\bf{X}
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$$
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## Problem 2
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Please find the general solution of
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$$
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\bf{\dot{X}} =
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\begin{bmatrix}
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-6 & 5 \\
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-5 & 4 \\
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\end{bmatrix}
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\bf{X}
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$$
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# Python Problems
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## Problem 3
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The Archimedes Spiral can be plotted by taking all the positive whole numbers (e.g.j = 0, 1, 2, 3, 4, 5, ...) and putting them into the format $n = (j,j)$ , and plotting them in polar coordinates where the first term, $n_1$, is the radius, and the second term, $n_2$, is the angle in radians.
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### Part A
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You need to plot the first 1000 terms in a scatter plot. In addition, we would like to only look at the top right quadrant! What you're going for is shown in Figure 1.
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### Part B
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You need to plot the first 25 terms, looking at th eentire polar plot (all quadrants, and then, put a *smooth* line through it. What you're going for is shown in Figure 2.)
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Hint: [This will be a useful reference](https://matplotlib.org/stable/gallery/pie_and_polar_charts/index.html)
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## Problem 4
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Consider the following system:
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$$
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\bf{\dot{X}} =
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\begin{bmatrix}
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1 & 2 & 1\\
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3 & 1+x & 1\\
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1 & 0 & 0
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\end{bmatrix}
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\bf{X}
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$$
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This linear differential equation system’s behavior is governed by its eigenvalues. In particular, the eigenvalues relate to stability and we may wish to see where they cross the 0 line (in terms of their real value). The constant x varies over the interval [−5, 5]. Using a Jupyter Notebook (local, or on Google Colab), Python, NumPy, and Matplotlib’s PyPlot, you should evaluate the eigenvalues for 50 evenly spaced values of x between −5 and 5, and produce a plot that visualizes the variation in the three eigenvalues as x varies. An example plot is shown in Figure 3 (for a different matrix!)
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---
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**Documentation**
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- [x] ME2016-HW1 📅 2024-09-18 ✅ 2024-09-18
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- [x] Problem 1 ⏳ 2024-09-16 ✅ 2024-09-18
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- [x] Problem 2 ⏳ 2024-09-16 ✅ 2024-09-18
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- [x] Problem 3 ✅ 2024-09-18
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- [x] Part A ⏳ 2024-09-16 ✅ 2024-09-18
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- [x] Part B ⏳ 2024-09-16 ✅ 2024-09-18
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- [x] Problem 4 ✅ 2024-09-18
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- [x] Part B ⏳ 2024-09-16 ✅ 2024-09-18
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