## Course material for M231, Jochen Denzler

We used the textbook by Nagle, Saff, Snider: Fundamentals of Differential Equations (5th ed).

I set up the following notes to complement the textbook with more conceptually focused material. I make them available with the hope that others may find them useful.

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.

I have no pretense that they are a substitute for the textbook.

notesCh1.ps, notesCh1.pdf
3 1/2 pages on concepts and basic facts
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notesCh2.ps, notesCh2.pdf
5 pages on different types of 1st order equations. You won't find exact differential equations here, because I consider them as a footnote to vector calculus. But I do include the energy integral from Newton's equations as a different illustration for the idea of an integrating factor.
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notesCh3.ps, notesCh3.pdf
In the modelling section, the textbook(s) have/has so many brine problems that one could think the Great Salt Lake came into existence when an experimental ODE class ran out of control. Here is a challenge problem that is pure modelling (no solution technique); I did a lot of individual tutoring with this one. I have a solution, but won't make it available lest this problem contract the brinomania, too.
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notesCh4.ps, notesCh4.pdf
This is on 2nd (and higher) order linear equations, and here I deviate significantly from the usual textbook approach. I first lay out a roadmap (aka table of contents) for solution techniques; then I stress that the methods work for n-th order as well as for 2nd and have much more to do with linearity than ODEs proper. Otherwise, the contrary claim would be absorbed by the students tacitly. -- I am adamant about including the case n=1, such as to relate the new material with the old on 1st order. And I stress the role of complex numbers. The rule ``You try erx and if you find r=2+3i or 2-3i, then you abandon our attempt and declare e2x(a cos 3x + b sin 3x) a solution'' is an offense to the brain. So I make Euler's formula into core material, not auxiliary material.
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notesCh4a.ps, notesCh4a.pdf
The free and forced oscillator:
We can't do everything: I omit the critical and overdamped case in favor of a really thorough discussion of the case of small damping, including a careful discussion of resonance and phase shifts, and their qualitative interpretation, and the convenience of complex numbers in practical calculations here. I expect my students to recognize the resonance phenomenon when they encounter it in the real world. That's tough on them, but it would be even tougher to require their attention while pretending it's not meant to be used outside the classroom.
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notesCh7.ps, notesCh7.pdf
On Laplace transforms
These notes are regretfully incomplete, because, near the end of the semester, I couldn't include convolution nor delta impulses. I still hope they may be useful to get the gist of the method even from the introductory secions on.

No, I did not include a section on partial fraction decomposition in the notes, fine-tuned to the new context. Calculus notes on the subject are available, and students must adapt to the somewhat different notation and context. (Yes, we did repeat it in class.)

Link to those of my calculus notes concerning (integration by) partial fractions:
pdf file or dvi file or ps file. Textbook references therein refer to Thomas Calculus ed 10, used in the course for which I prepared these notes originally.