Here I shall try to keep an updated account of what has been done so far in the course.

date	what we did	comments
28/3	introduction to the course first order scalar equations, some special cases (example of non-uniqueness and explosinon of solutions) equations with separable variables	page 1-7 of my notes (see main page)
29/3	more on separable equations: uniqueness homogeneous equations linear equations (scalar, first order)	page 8-13 of my notes
31/3	The Bernoulli and Riccati equations The general initial value problem Lipschitz conditions, uniqueness
4/4	The existence theorem Extension of solutions	for 31/3 and 4/4: page 14-24 in the notes
5/4	The lab: how to use ode45 How to make systems out of higher order equations Numeical methods, Euler (with convegence theorem, not yet the proof), Heun, Runge Kutta
7/4	cancelled
25/4	A very short recollection of what we did before Easter break Maximal solutions: the theorem and the proof Differential inequalities and the Gronwall lemma
26/4	Exercises Introduction to linear systems: the existence proof.	The proof of Lipschitz continuity was not very clear ... an explanation can be found in this note
28/4	Linear algebra Linear systems: three equivalent statements about solutions to systems of ode's
2/5	Results around the (very important) statement that the set of solutions to a system of ode's form an n-dimensional subspace of C1. The fundamental solution, and the variation of constants formula
3/5	Stability and continuous dependence on data and small perturbations of a linear system Systems with constant coefficients. The matrix exponential and some properties of functions of matrices	Corresponds to pages up to 56 in the notes
9/5	The Cayley-Hamilton theorem and the evaluation of analytical functions of a matrix Fundamental solutions for linear equations of higher order.
10/5	Notions of stability of solutions to differential equations A stability theorem for solutions to equations of the form y' = A y + g(t,y)
12/5	Proof of an instability theorem for equations of the type y' = A y + g(t,y)
16/5	Lyapunov functions
17/5	Lyapunov-functions, an example Two point boundary value problems Fundamental solutions
19/5	computer lab