---
title: Homework 1
allDay: true
date: 2024-09-18
endDate: null
completed: 2024-09-26T09:55:06.474-04:00
type: single
startTime: 10:00
endTime: 12:00
---
# Instructions
#Homework
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.

Please upload a Jupyter Notebook for problems 3 and 4.

Problems 1 and 2 are worth 10 points each, problems 3 and 4 are worth 15 points each.

# Written Problems
## Problem 1
Please find the general solution of 
$$
\bf{\dot{X}} = 
\begin{bmatrix}   
-1 & 5 & 2\\   
4 & -1 & -2\\
0 & 0 & 6
\end{bmatrix}
\bf{X}
$$

## Problem 2
Please find the general solution of 
$$
\bf{\dot{X}} = 
\begin{bmatrix}   
-6 & 5 \\   
-5 & 4 \\
\end{bmatrix}
\bf{X}
$$
# Python Problems
## Problem 3
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.
### Part A
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.
### Part B
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.)
Hint: [This will be a useful reference](https://matplotlib.org/stable/gallery/pie_and_polar_charts/index.html)
## Problem 4
Consider the following system:
$$
\bf{\dot{X}} = 
\begin{bmatrix}   
1 & 2 & 1\\   
3 & 1+x & 1\\
1 & 0 & 0
\end{bmatrix}
\bf{X}
$$
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!)

--- 
**Documentation**
- [x] ME2016-HW1 📅 2024-09-18 ✅ 2024-09-18
	- [x] Problem 1 ⏳ 2024-09-16 ✅ 2024-09-18
	- [x] Problem 2 ⏳ 2024-09-16 ✅ 2024-09-18
	- [x] Problem 3 ✅ 2024-09-18
		- [x] Part A ⏳ 2024-09-16 ✅ 2024-09-18
		- [x] Part B ⏳ 2024-09-16 ✅ 2024-09-18
	- [x] Problem 4 ✅ 2024-09-18
		- [x] Part B ⏳ 2024-09-16 ✅ 2024-09-18
