## Course material for M431, Jochen Denzler

I decided to go this one altogether without a textbook (often we use
Boyce-diPrima). So the notes and homework represent the course fairly well.
But as always with first time notes, they may still deserve some polishing;
they were created as the course went. --- They may be helpful as
complementary reading also for a textbook-oriented course.
**I reserve the copyright in all course material created by me and made
available here. You are welcome to use it for teaching, learning, or
research as long as you are not using it commercially.**

### First Subject: The power series method

An auto-plagiarized calculus manuscript on power series, slightly adapted,
to repeat power series (possibly to understand'em the first time, as they
often get squeezed in unlovingly at the end of a Calc' course):

PowerSeries.dvi,
PowerSeries.ps,
PowerSeries.pdf

---

Homework sheet number 1:

Homework1.dvi,
Homework1.ps,
Homework1.pdf

---

Power series method (Frobenius method) for ODEs:

Frobenius.ps,
Frobenius.pdf

---

Homework sheet number 2:

Homework2.dvi,
Homework2.ps,
Homework2.pdf

---

Homework sheet number 3:

Homework3.dvi,
Homework3.ps,
Homework3.pdf

### Second Subject: First Order Systems; Phase Space

An introduction to systems of ODEs; examples; concept of phase space;
conserved quantities:

systems.ps,
systems.pdf

---

Vectors and matrices, Linear systems, Exponentials of matrices;
Stability of equilibria, linear and nonlinear (basic facts, and intuition,
but no formal proofs -- many students came from engineering!):

linsys.ps,
linsys.pdf

---

Homework sheet number 4:
Homework4.dvi,
Homework4.ps,
Homework4.pdf

Homework sheet number 5:
Homework5.dvi,
Homework5.ps,
Homework5.pdf

Homework sheet number 6:
Homework6.dvi,
Homework6.ps,
Homework6.pdf

---

**Missing material: **alas no time left for the matrix version of variation
of parameters; but with the thorough study of matrix exponentials, students
should be able to adapt the M231-method to matrices quickly, when they
actually need it. As many in the audience will not have another ODE course,
I felt it more important to offer a glimpse of the geometric insight that is
so paramount to a contemporary understanding of ODEs. They would have more
difficulties studying this on their own. --- Alas also no time left for
coupled oscillations. Next time, I'll streamline the material to fit it in...