MAT 665 Homework, Spring 2026

Note: Whenever convenient, you can use let \({\mathbf x} = (x_1, x_2, x_3)\) or \({\mathbf x} = (x, y, z)\). Be sure to draw an arrow over a vector. The first form can be awkward with an initial condition, so the usual notation is \({\mathbf x}(0) = (x_0, y_0, z_0)\)

Due Thursday, January 22, at 11:59pm (Week 2).
WeBWorK

Due Friday, January 30 (Week 3), in class.
Assignment 2
Problem Set 1.1: 1acde, 2ac, 3, 4, 5.
Use Swift’s convention of single and double (and possibly triple) arrows for all the phase portraits.

Extra credit: In problem 3, I suggest you draw the phase portrait for 7 valus of \(a\), not just the three mentioned: \(a \in \{-2, -1, -1/2, 0, 1/2, 1, 2\}\). Do these problems quickly. You don't have to write all the solutions, just sketch the phase portraits using single and double arrows to guide you.

Bonus 1: Eliminate \(t\) in the general solution to \(\frac{dx}{dt} = -x, \frac{dy}{dt} = 2y\). Sketch the 2-D phase portrait fairly accurately.

Bonus 2: Sketch the 2-D phase portraits for \(\frac{dx}{dt} = -x, \frac{dy}{dt} = -2y, \frac{dz}{dt} = 3z\) within each of the coordinate planes, and attempt to sketch the 3-D phase portrait, as in these three-dimensional phase portraits.

Bonus 3: Try to find the general solution to \(y' = 2x y^2\) in the style of the general solution to \(y' = y^3\) derived in class on Jan. 16. See the desmos graph y' = y^3 general solution.
Hint: in the Bonus 3 problem, I found it useful to use \(\text{tanh}^{-1}(c)\), which is a bijection mapping \( (-1,1)\) to \( (-\infty, \infty) \).

Solutions: Here are the solutions to Assignment 2, and here are some Desmos graphs to accompany them:
Perko 1.1, problem 1(e),
Perko 1.1, problem 2(c),
Perko 1.1, problem 3,
y' = 2x y^2 general solution.

Due Friday, February 6 (Week 4), in class.
Assignment 3:
Problem Set 1.2: 1ac, 2, 3ac, 4, 5, 7.
Notes:
Do problem 1ac as written, using the Corollary. In the rest of the problems you can avoid the corollary as described below.
In problem 2, you can find the general solution the easy way, without the corollary. (No inverse is required.) Also, sketch the phase portrait in the \({\bf y}\) coordinates only.
In problem 4, you can use the general solution (with \({\bf c}\) and no matrix inverses) you found for 1(a) and 2, and find the value of \({\bf c}\) for the initial condition given by solving \(P {\bf c} = {\bf x}_0\) for \({\bf c}\).
In problem 5, do not give a proof. Just state the answer.
In 7d and 7e, assume \(|\lambda| < |\mu|\).

Bonus 1: Find the matrix A such that the ODE dx/dt = A x has the general solution x(t) = [1 0]^T e^-t c₁ + [1 1]^T e^-2t c₂
(In problem 5, Perko uses the notation \( (1, 0)^T\) where I write \([1 0]^T\).)

Bonus 2: Find a 2x2 non-diagonal matrix \(A\) whose eigenvalues are 2 and 3, and sketch the phase portrait of dx/dt = A x.
Solutions to Assignment 3.

Due Friday, Feb. 13 (Week 5) in class
Assignment 4
1.3: 1b*, 5, 6, 7, 8 (you can use propositions from this section.)
1.4: 1, 2, 5*, 6

Hints:
*In 1.3 #1(b) Use this desmos graph to plot the set \(\{ A {\bf x} : |{\bf x}| = 1 \}\). Estimate roughly (say, to the nearest tenth) the minimum and maximum value of \( |A {\bf x}|\) on this set. Write down the function \(f(\theta) = |A(\cos(\theta), \sin(\theta))|\). By definition, \(\|A\|\) is the maximum value of \(f\). The calculus for finding the max of \(f\) is rather involved, so do not use Calculus I techniques. Rather, follow the hint in the book. Either use Lagrange multipliers to solve the optimized constraint problem, or use the fact that \(\|A\| = \sqrt{\text{Max eigenvalue of }A^TA}\). If you use this second method, verify your answer by plotting \(y = f(\theta)\) with Desmos, and use Desmos to estimate the max and min values of \(f\). Also verify that the min value of \(f\) is the square root of the smaller eigenvalue of \(A^TA\). Make a sketch of the graph of \(f\), or send me a link to the desmos graph using the "Share Graph" feature. Here is my graph for problem 1(c), which is not assigned.
* You don't need to invert any matrices to do proplem 5(c). You would need to compute P^-1 to compute e^At as in section 1.2, but it's OK to find the general solution in a form that does not require P^-1. Alternatively, you can sum the power series for \(\exp(At)\).

Due Wednesday, February. 18 (Week 6) in class
Assignment 5
1.5: 1, 3, 5, 7

Due Monday, March 2 (week 8) in class
Assignment 6
1.6: 1-4, along with prolems H1 and H2 at the end of my handout on complex eigenvalues.

Notes for problems from section 1.6, referring to the equation numbers in the handout.
In problem 1, solve the initial value problem using both Equation (4) and Equation (7). Make an estimate of the ratio of the times it takes to do the two methods.
On problems 2-4, you don't have to solve the IVP. Find the general solution using Equation (3), like I did in class. (You don't need to invert any matrices.)
On problem 2, sketch the phase portrait in the \(x_1 - x_2\) plane fairly accurately, using my technique. Then, attempt to draw a 3-D phase portrait.
Problems 3 and 4 very are similar to the two examples at the bottom of page 2 in my handout.

Here are the solutions to Assignment 6 from Perko section 1.6
Here is a Mathematica noteboook to check solutions to problems in Perko section 1.6.
Here are solutions to Assignment 6, problems H1 and H2. Here are the darker black and white solutions to H1 and H2.

Due Monday, March 30. (Week 11)
Assignment 7
1.7: Problems 1 a b c d, 2 a c d in the book.
Let \( A = \begin{bmatrix}-2 & 2 & -1 \\ -7 & 5 & -2 \\ 5 & -3 & 3 \end{bmatrix} \). It is not obvious, but the eigenvalues of \(A\) are \(\lambda = 2, 2, 2\). Solve the ODE (1), namely \(\dot{\bf x} = A {\bf x}\), \({\bf x}(0) = {\bf x}_0\), with
(a) \({\bf x}_0 = (1, 0, 0) \)
(b) \({\bf x}_0 = (1, 3, 0) \)
(c) \({\bf x}_0 = (1, 1, -2) \) Note: This was corrected at 1:04pm on 2026/03/23.
(d) The general \({\bf x}_0\). That is, evaluate \(\exp(At)\).
Note added on 2026/03/27: In parts (a), (b), and (c) use the formulas on our home page involving \({\bf x}_0\), \({\bf x}_1\), and \({\bf x}_2\). In part (d) use that formula on our home page (which is the last equation in Perko, pg. 33).

Due Wednesday, April 8 (Week 12)
Assignment 8
Section 1.8: 5*, 6*, 7.
In problem 5b, choose 3 of the 7 possible cases, and write down the form of the general solution (like I did in class) for them.
New In problem 5, follow this example of my solution to problem 3.
In problem 6, write down the general solution but do not solve the IVP for general \({\bf x}_0\). Do not invert any matrices! You may use Mathematica or your calculator to do the row reductions for problems 6 in the book, but you can probably do them by hand.

Solve the single ODE (D-1)³ x = x''' - 3 x'' + 3x' - x = 0  for x(t) in two ways:
(a) Find the general solution using MAT 239 techniques.
(b) Convert the single ODE to a system x dot = Ax, where A is a 3x3 matrix, and find the general solution using the JCF. Note that the scalar x is the first component of x.
(c) Explain (but don't formally prove) how the algorithm we learned in MAT 239 implies that the matrix A obtained for any single linear homogeneous ODE with constant coefficients has a Jordan Canonical Form with only one block for each eigenvalue.

I will also send you email with a matlab file. You will do the problems and reply with your solutions attached.