Fall 2024: Differential Equations II (Math 332)

Overview

Lecture sketches and homework

To help you study for this class, I am posting lecture sketches. These are not a substitute for your personal lecture notes or coming to class (for instance, lots of details and motivation are not included in the sketches). I hope that they are useful to you for revisiting the material and for preparing for exams.

About the homework

• Homework problems are posted for each unit. Homework is submitted online, and you have an unlimited number of attempts. Only the best score is used for your grade.

  Most problems have a random component (which allows you to continue practicing throughout the semester without putting your scores at risk).

• Aim to complete the problems well before the posted due date.

  A 15% penalty applies if homework is submitted late.

• Collect a bonus point for each mathematical typo you find in the lecture notes (that is not yet fixed online), or by reporting mistakes in the homework system. Each bonus point is worth 1% towards a midterm exam.

  The homework system is written by myself in the hope that you find it beneficial. Please help make it as useful as possible by letting me know about any issues!

Exams and practice material

The following material will help you prepare for the exams.

• Midterm Exam 1:
  midterm01-practice.pdf, midterm01-practice-solution.pdf
  midterm01.pdf, midterm01-solution.pdf
• Midterm Exam 2:
  midterm02-practice.pdf, midterm02-practice-solution.pdf
  midterm02.pdf, midterm02-solution.pdf

Sage

For more involved calculations, we will explore the open-source free computer algebra system Sage.

If you just want to run a handful quick computations (without saving your work), you can use the text box below.

A convenient way to use Sage more seriously is https://cocalc.com. This free cloud service does not require you to install anything, and you can access your files and computations from any computer as long as you have internet. To do computations, once you are logged in and inside a project, you will need to create a "Sage notebook" as a new file.

Here are some things to try:

• Sage can solve certain differential equations. For instance, to solve Example 17 in Lecture 3:

  y = function('y')(x)
  desolve(diff(y,x,2) - diff(y,x) - 2*y == 0, y)
  

• Sage can solve (symbolically!) simple initial value problems. For instance, to solve Example 18 in Lecture 3:

  y = function('y')(x)
  desolve(diff(y,x,2) - diff(y,x) - 2*y == 0, y, ics=[0,4,5])
  

  Unfortunately, this code currently only works for differential equations of first and second order. To likewise solve a third-order differential equation, we can use the function desolve_laplace instead:

  y = function('y')(x)
  desolve_laplace(diff(y,x,3) == 3*diff(y,x,2) - 4*y, y, ics=[0,1,-2,3])
  

  The output shows that the function $ y(x) = \frac13(3x - 2) e^{2x} + \frac53 e^{-x} $ is the unique solution to the initial value problem $ y''' = 3 y'' -4 y $ with $ y(0) = 1 $, $ y' (0) = -2 $ and $ y'' (0) = 3 $.
• Sage is happy to compute eigenvalues and eigenvectors. For instance, to solve Example 54 in Lecture 8:

  A = matrix([[8,-10],[5,-7]])
  A.eigenvectors_right()
  

• Sage can even compute general powers of a matrix. For instance, to solve Example 62 in Lecture 10:

  n = var('n')
  M = matrix([[0,1],[2,1]])
  M^n
  

• Similarly, Sage can compute matrix exponentials. For instance, to solve Example 70 in Lecture 11:

  M = matrix([[8,-10],[5,-7]])
  exp(M*x)
  

  (Note that x is predefined as a symbolic variable in Sage; that's why we don't need x = var('x') as in the case of n above.)
• We can plot phase portraits as follows:

  x,y = var('x y');
  streamline_plot((x*(y-1),y*(x-1)), (x,-3,3), (y,-3,3))
  

• Using matrix exponentials, we can easily solve inhomogeneous systems of differential equations. For instance, to solve Example 97 in Lecture 19:

  t = var('t')
  A = matrix([[2,3],[2,1]])
  y = exp(A*t)*integrate(exp(-A*t)*vector([0,-2*e^(3*t)]), t)
  y.simplify_full()
  

  Or, to solve Example 98(c) in Lecture 20:

  s, t = var('s, t')
  A = matrix([[1,2],[-1,4]])
  y = exp(A*t)*vector([1,2]) + exp(A*t)*integrate(exp(-A*s)*vector([0,2*e^s]), s,0,t)
  y.simplify_full()
  

• We can also plot functions. For instance, the following are the first terms of the Fourier series we compute in Example 137 in Lecture 27:

  t = var('t')
  plot(4/pi*(sin(t) + 1/3*sin(3*t) + 1/5*sin(5*t)), (t,-pi,pi))
  

  To include more terms, it is convenient to use the sum command:

  t = var('t')
  plot(4/pi*(sum(sin(n*t)/n for n in range(1,19,2))), (t,-pi,pi))
  

• The following plots the solution to the heat equation with initial temperature profile equal to 1 and boundary conditions of zero on both ends.

  x, t = var('x,t')
  uxt = sum(4/(pi*n) * exp(-pi^2*n^2*t) * sin(pi*n*x) for n in range(1,20,2))
  density_plot(uxt, (x,0,1), (t,0,0.4), plot_points=200, cmap='hot')