Due Jan 30 before 4pm
Chapter 1: #8,11ab,12,14
Chapter 5: #11,
from class: prove the properties of the determinant using elementary matrices (changes sign if two rows or columns are exchanged; does not change if a multiple of a row or columns is added to a different row or column; is multipled by a number is a row or column is multiplied by a number),
from this web site please do #1,2,3,4 from the "Matrix algebra review; Eigenvalues and eigenvectors" section http://www.math.wisc.edu/~angenent/519.2016s/notes/math519problems.html#ev-ev

Due Feb 6 before 4pm
Chapter 5: end the proof from class; #5cf,6,9,10
Chapter 6: #13,14

Due Feb 13 before 4pm
From the web page http://www.math.wisc.edu/~angenent/519.2016s/notes/math519problems.html#exptA do problems #2,4,7,10,11.
Chapter 6: #1cfg,

Due Feb 27 before 4pm
From the Web page http://www.math.wisc.edu/~angenent/519.2016s/notes/math519problems.html#existence-uniqueness do problems #2,3,4,6,7,9abc
Show that in the proof of the existence of solutions we have |x_k(s) - x(s)|<= L\delta (1/2)^{k-1}

Due March 6 before 4pm
Assume $x$ is differentiable on $[t_0, T)$. Prove that if $\lim_{t\to T} x(t)$ and $\lim_{t\to T} x'(t)$ exist, then $x$ is differentiable on $[t_0, T]$. (Hint: use the Mean Value Theorem.)
From http://www.math.wisc.edu/~angenent/519.2016s/notes/math519problems.html#variationalEquation do #1abd.
Calculate the first two terms (order zero and one) of the $\alpha$-Taylor expansion of the solution of the IVP x' = x(1-x)+$\alpha$ sint, x(0)=0, as described in class.
From Chapter 7 do #1bce,2,5,6

Due March 13 before 4pm
From Chapter 8 do #1,2,5ab,10
From class: Let X' = F(X) be a differential equation in the plane (X = (x,y)), and F(p) = 0 an equilibrium. If DF(p) has one complex eigenvalue (and its conjugate) with negative real part, and if X_0 is close enough to p, then prove that X(t,X_0) approaches 0 as t goes to infinity.

Due April 3 before 4pm
Chapter 8: #3,4,5,6,9,11,12

Due April 10 before 4pm
Prove that if a system have a Lyapunov function, then all sets U_a={x, such that V(x) <= a} are invariant under the flow.
Can you use linerization to prove that the system
x' = -x+y+yx
y'=x-y-x^2-y^3
has an asymptotically stable equilibrium at the origin? Use a Liapunov function to prove it.
Same question for the harmonic oscillator x'' + sinx=0, and for x''+kx = 0, k>0.

Due April 17 before 4pm
Problems 1,2,3,4,5,8 in the page http://www.math.wisc.edu/~angenent/519.2016s/notes/math519problems.html#Poincare-maps

Due May 4 before 4pm
#9 from http://www.math.wisc.edu/~angenent/519.2016s/notes/math519problems.html#Poincare-maps
Consider the equation x' = x(4+ sin t - x^2) (very similar to the one in the midterm). Answer the questions in the midterm for this new equation and prove that there exists a periodic solution with initial condition between 0 and 5.
Prove that the Poincare map on the plane is monotone using a topological argument, the way we did in class.
From Poincare-Bendixson Theorem, prove that if Y is in w(X), then w(Y) is not void.
Chapter 10: #7, 8(a) and 10

*************************************