ORDINARY DIFFERENTIAL EQUATIONS
(+ Oscillations & Waves)
PAPER CP3/CP4, FIRST YEAR
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Prof Alexander Schekochihin
 A sketch of students (or, perhaps, fellows) in a manuscript of William of Ockham's commentary on Aristotle's Physics (MS293 from the Merton College library, image courtesy of J. Walwarth). |
Michaelmas Term 2021 LECTURE NOTES (containing also the reading suggestions and the problem sets) --- this document may be corrected/amended as we proceed; I will be very grateful for your feedback: comments, error corrections, views etc. This version is of 10.10.21. Check back here for updates! LECTURES Monday 12:00-13:00 (week 2) Tuesday 12:00-13:00 (week 2) Wednesday 12:00-13:00 (week 2) Monday 11:00-12:00 (weeks 6, 7) Tuesday 10:00-12:00 (weeks 6, 7) Monday 10:00-12:00 (week 8) Tuesday 10:00-11:00 (week 8) in Martin Wood LT except on Wednesday week 2, when we will meet Lindemann LT |
A sketch of students (or, perhaps, fellows) in a manuscript of William of Ockham's commentary on Aristotle's Physics (MS293 from the Merton College library, image courtesy of J. Walwarth). |
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| Part I: GENERAL INTRO; FIRST-ORDER ODEs |
Lecture 1 (12:00-13:00; Monday
18.10.21) Language of the game: what is an
ODE? integral curves and the Cauchy problem, existence
and uniqueness.
READING:
Lecture Notes
Sec 1.1-1.3
Lecture 2 (12:00-13:00; Tuesday
19.10.21) ODEs in symmetric form, parametric
solutions. Methods for solving first-order ODEs.
READING:
Lecture Notes
Sec 1.4, 2
Lecture
3 (12:00-13:00;
Wednesday 20.10.21 in
Lindemann LT) Methods for
solving first-order ODEs cont'd.
You are ready to do Problem Set 1
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| Part II: LINEAR/SECOND-ORDER ODEs |
Lecture 4 (11:00-12:00; Monday
15.11.21) Language of the game cont'd:
higher-order ODEs, systems of ODEs, phase space and
phase portrait. The nonlinear pendulum.
READING:
Lecture Notes
Sec 3.1-3.5.2
Lectures 5-6 (10:00-12:00; Tuesday
16.11.21) Linear ODEs. Existence and
uniqueness. Superposition principle. Second-order
linear ODE with constant coefficients: homogeneous
equation (non-degenerate case). Damped oscillator.
General scheme for solving homogeneous linear
equations. Strategy for determining the fundamental
system for 2nd-order equations. Second-order linear
ODE with constant coefficients: homogeneous equation
(degenerate case). Other examples.
READING:
Lecture Notes
4.1, 4.2, 5.1-5.1.1, 4.3, 4.6, 5.1.2, 4.6.1
You are ready to do Problem Set 2
Lecture 7 (11:00-12:00; Monday
22.11.21) Linear ODEs: general scheme for
solving inhomogeneous equations. Second-order linear
ODE with constant coefficients: inhomogeneous
equations.
READING:
Lecture Notes
Sec 4.2.1, 4.4, 4.6, 5.2
Lecture 8 (10:00-11:00; Tuesday
23.11.21) Forced oscillator.
READING: Lecture
Notes Sec 5.3You are ready to do Problem Set 3
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| Part III: SYSTEMS of ODEs |
Lecture 9 (11:00-12:00; Tuesday
23.11.21) Systems of linear ODEs with
constant coefficients: diagonalisable systems with no
degenerate eignenvalues.
READING:
Lecture Notes
Sec 6.1
Lectures 10-11 (10:00-12:00; Monday
29.11.21) Systems of linear ODEs with
constant coefficients: Hermitian vs. non-Hermitian
systems. Schur's triangulation theorem.
READING:
Lecture Notes
Sec 6.2-6.3
Qualitative solution of autonomous ODEs: Poincare's classification of 2D equilibria. READING:Lecture
Notes Sec 8.1
Lecture 12 (11:00-12:00; Tuesday
30.12.21) Poincare's classification of 2D
equilibria cont'd.
You are ready to do Problem Sets 4 & 5 (except Q5.3 & 5.4) |
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This course will
continue in HT and cover normal modes in systems of
"masses on springs" and the like.
ODE Lecture Notes contain the material on normal modes (Chapter 7) and its accompanying problem set (PS-6). The follow-up course, on the Wave Equation and its solutions, will be taught by Prof Matt Jarvis. His lecture notes are here. Last year's notes by Prof Felix Parra are here. LECTURES Wednesday 19 January 10:00-12:00 in Lindemann LT Monday 24 & 31 January 11:00-12:00 Tuesday 25 January 11:00-12:00 in M. Wood LT |
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| Part IV: LINEAR OSCILLATIONS |
Lecture 13 (10:00-11:00; Wednesday
19.01.22) Normal modes: coupled identical
oscillators.
Lecture 14 (11:00-12:00; Wednesday
19.01.22) Energy of oscillators. Damped
oscillators. Forced oscillators.
Lecture 15 (11:00-12:00; Monday
24.01.22) Unequal oscillators.
READING:
Lecture Notes
Sec 7.3.1, 7.3.2, 7.3.5
You are ready to do Problem Set 6: Q6.3-6.6 |
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| Part V: NONLINEAR OSCILLATIONS |
Lecture 16 (11:00-12:00; Tuesday
25.01.22) N coupled oscillators and the
emergence of the wave equation. Limit cycles.
Auto-oscillations.
Lecture 17 (11:00-12:00; Monday
31.01.22) Relaxation oscillations: overdamped
van der Pol oscillator. Population dynamics: Odell's
model (Q5.4 of PS-5). READING:
Lecture Notes
Sec 8.4.2
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Trinity Term 2022 REVISION LECTURES in weeks 1-3 Lecture 18 (11:00-12:00; Tuesday
26.04.22) First-order ODEs.
Lecture 19 (11:00-12:00; Monday 2.05.22) First-order ODEs cont'd. Linear ODEs. Lecture 20 (10:00-11:00; Monday 9.05.22) Linear ODEs: 2nd order. Systems of linear ODEs. |
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