MT1222: Calculus and Applications -- Part 1: ODEs

This first half of the course, dealing with ordinary differential equations (ODEs), is taught by Dr. Matthias Heil. The second half, dealing with mechanics, is taught by Dr. Rich Hewitt who provides a separate page for his course notes. This page provides online access to the lecture notes, example sheets and other handouts and announcements. Most of the material will be taught in "chalk and talk" mode. If OHP transparencies are used, copies will be made available (after the lecture) on this page.

Please note that the lecture notes only summarise the main results and will generally be handed out after the material has been covered in the lecture. You are expected take notes during the classes.

The Riot Act -- Please Read
The lecture course (four hours a week) is accompanied by weekly supervision classes during which you will get a chance to discuss the material in small groups, and to resolve any problems. The course is a "methods course" and I can assure you that the only way to understand the course material is to work your way through LOTS AND LOTS of examples. Suitable example sheets will be made available (in pdf format) on this page. You MUST attempt the questions on these sheets and hand them in before your weekly supervision. (Your supervisor will arrange details with you.) He/she will mark the most important questions (as identified by me) and give you feedback during the supervision.

The supervisions are for YOUR benefit -- if you have any problems with the material, YOU have to take the initiative and raise them during your supervision class. You will be expected to have at least tried to solve the problem so that you are in a position to ask specific questions. Most of the problems are extensions/modifications of examples presented in class, so it's a good idea to have a look through your lecture notes before attempting the questions on the example sheets. The following exchange tends to take place at the beginning of most of my lecture courses:

Student: "I can't do that question"
Lecturer: "What exactly is the problem? How far did you get?"
Student: "I don't even know how to start"
Lecturer: "Well, this is essentially the same example that we did in class; have you looked at your lecture notes?"
Student: "No, I can't find them" [or "No, I didn't go to the lecture", "No, why should I?", etc...].
Lecturer: "[censored]".

To be absolutely clear: If you can't do a question even though you've tried really hard [see below for a definition of "really hard"], and you have consulted all the available material, I'll be delighted to help you -- that's my job, after all. However, if you expect me (or my colleagues) to turn supervisions into repeat performances of the lectures because attending them would interfere too much with your social life, or if you can't be bothered to read the handouts, I will have very little sympathy.

I will distribute detailed solutions (again in pdf format on this page) after the material has been covered in the supervisions but these should only be used to check your solutions. Do not make the mistake of assuming that it'll be sufficient to look at somebody else's solution to understand the material! "Maths is not a spectator sport!".

While I'm having a general rant, let me address another frequently-asked-question:

How much time should you expect to spend on this course? As I said above, you are expected to attend the lectures and the weekly supervision classes. However, you will obviously have to dedicate a considerable amount of additional time to understanding the material that's been covered in the lecture. As a rough guide, I'd say that the absolute superstars amongst you (if in doubt, assume that this does not include you) will need about an hour of private study per lecture to really understand the material, and to work your way through the example sheets; two hours is likely to be more realistic for the mere mortals amongst you. If you decide to cut corners you will pay the price, not only in this year's exam but also in all subsequent ones, because second- and third-year courses build on the material taught in this course.

Syllabus:

General introduction. Notation. What are ODEs? Implicit vs. explicit form. Classification: Order, linearity, autonomous ODEs. Boundary and initial conditions. Boundary and initial value problems. Existence and uniqueness for linear and nonlinear ODEs.
First order ODEs: Graphical methods; separable ODEs; ODEs of homogeneous type; integrating factor.
Second-order ODEs: Existence and uniqueness. Linear ODEs: superposition of solutions, fundamental solutions and the general solution for homogeneous ODEs. The general solution of constant-coefficient ODEs; particular solutions for specific RHS; the method of undetermined coefficients. [If time permits (probably not): Power series expansions about regular points.] Some nonlinear ODEs with special properties (autonomous ODEs and ODEs that do not contain the dependent variable).
Mechanics applications of second-order ODEs: Damped harmonic motions of mechanical oscillators; harmonic forcing and resonance.
Non-dimensionalisation and scaling. Exploiting small parameters in an ODE: perturbation methods. Motivation via the roots of quadratic polynomials (singular perturbations only mentioned); application to selected (regularly perturbed) ODEs.

Material for supervisions:

Since this course is being given for the first time, it will be difficult to predict exactly what material will have been covered at a specific point in time. "Synchronising" the example sheets with the lecture is therefore slightly non-trivial, especially since supervisions are being held throughout the week. As a result, you may occasionally come across a question that deals with material that has not been covered in the lecture. Since this tends to cause instant panic amongst students, I propose the following procedure: Example sheets will be made available as soon as possible and you should check this webpage frequently for any new material. I will update the box below after every lecture to state which questions I regard as "do-able". Being keen students, you will then immediately start to work on these to make sure that you can bombard your supervisor with questions if you encounter any problems.

Example Sheets:

Example Sheet 1 (pdf) .
Example Sheet 2 (pdf) [Question 5b corrected 13/02/06].
Example Sheet 3 (pdf) .
Example Sheet 4 (pdf) .
- Please note the correction in the discussion of the final step of the method of undetermined coefficients on the handout for Chapter 4. This is relevant for Q1(c)iv [24/02/06].
- There's a correction for Q2(c).
Example Sheet 5 (pdf) .
Example Sheet 6 (pdf) . [Note the correction to a comment in Question 1 (the correction doesn't affect the question itself) 3/3/06].

Solutions to Example Sheets:

Do-able questions

Date	Topics covered	Do-able questions
30/01/06	Introduction; notation; classification (order, linearity; autonomous ODEs); examples of ODEs and solutions; motivation for existence and uniqueness; counter-example for existence; uniqueness and boundary/initial conditions; number of constraints related to order of ODE; formal definition for IC/BC and IVP/BVP; example for IVP (1D motion of a particle subject to a prescribed force); example for BVP (transverse deflection of a string under constant tension).	Example sheet 1 (all questions!)
03/02/06	Counterexample for uniqueness. "Proper" theory: Existence and uniqueness for first-order nonlinear ODEs; examples. Existence and uniqueness for linear first-order ODEs; examples. Graphical solutions for first-order ODEs: The direction field; integral curves. Graphical explanation for how non-uniqueness may arise at points where f(x,y) is discontinuous. Asymptotes, stability of solutions. Isoclines. y'=-x/y as an example of an ODE for which a unique solution exists for most initial values, but only for a limited range of x values.	Example sheet 2: Q1.
06/02/06	Further discussion of y'=-x/y; definition of critical points. Separable ODEs; two ways of incorporating initial conditions; examples. ODEs of homogeneous type; examples. First-order linear ODEs: Motivation for integrating factor.	Example sheet 2: Q1,2,3,4.
10/02/06	Finish off integrating factor; example. Observation: The solution of a linear ODE can be written as the sum of a particular solution of the full equation and the general solution of the homogenous ODE. Marvel at this for a while. 2nd-order ODEs. General statement of IVPs and BVPs. Specific theory for 2nd-order linear ODEs: Existence and uniqueness; the homogeneous ODE and the superposition of its solutions; linear (in)dependence of functions; fundamental solutions for homogeneous ODEs (they're not unique!); the general solution of the inhomogeneous ODE. Example; demonstrate that choosing different particular solutions and different fundamental solutions in the general solution does not change the solution of the IVP.	Example sheet 2: All questions. Example sheet 3: Q1&2.
13/02/06	Illustration that the structure of the general solution, x = x_P + x_H, occurs in many other contexts such as linear algebra. Summary of the solution procedure for linear ODEs: Fundamental solution of the homogeneous ODE; particular solution of the inhomogeneous ODE; the sum yields the general solution; BC/IC determine the arbitrary constants in the fundamental solution. Constant coefficient ODEs: exp(lambda x) for homogenous solutions: 3 cases: distinct real roots; repeated root; complex conjugate roots. Examples.	Example sheet 3: All questions.
17/02/06	Particlar solutions for constant-coefficient ODEs: The method of undetermined coefficients as a trial-and-error-method, guided by the form of the RHS. Examine the method and its pitfalls for exponential forcing. Interpret "pathological" cases in terms of roots of the characteristic polynomial and (alternatively) as cases in which the RHS is a solution of the homogeneous ODE. Generalise to arbitrary RHSs consisting of multiple, linear independent functions.	Example sheet 4: Q1a,b,c and Q2.
20/02/06	Finish off method of undetermined coefficients for general RHS including the modifications required (i) if derivatives of functions on the RHS create new, linearly independent functions when differentiated and (ii) if one of the functions on the RHS is a solution of the homogeneous ODE. Examples for all cases.	Example sheet 4: Everything apart from Q3.
24/02/06	Nonlinear 2nd order ODEs of special type: 2nd order ODEs that don't contain the dependent variable y''=f(x,y'). Autonomous ODEs y''=f(y,y'). Examples for both cases. Start "mechanics applications of second-order ODEs": Perform experiment with mechanical oscillator (mug on rubber string) and derive the governing equation.	Example sheet 4: All questions. Example sheet 5: Q1 and Q2.
27/02/06	Solutions of normalised "mechanical oscillator" equations. Interpretation of the four types of solutions of the homogeneous equation (pure damping; critical damping; damped oscillations; undamped oscillations). Interpretation of delta and omega as timescales for the decay of the oscillation and timescale of the undamped oscillations, respectively. Particular solution for harmonic forcing. Dependence of the amplitude of the response on various parameters (and their ratios!). Quasi-steady limit; high-frequency limit; large amplitudes for excitation near eigenfrequency. Resonance for delta=0 (mathematical symptom: forcing function is solution of the homogeneous ODE). Amplitude grows linearly.	Example sheet 5: All questions.
03/03/06	Motivation for perturbation methods: Forced mechanical oscillator with small forcing frequency. Argue heuristically that for Omega << 1, it should be possible to approximate the ODE m x'' + k x' + c x = F cos(Omega t) by c x = F cos(Omega t). Verify by examining the limit of the exact solution for small Omega. Algebraic example (roots of a second order polynomial) to illustrate the overall structure of perturbation expansions; motivated by form the of the Taylor expansion of the exact solution. Same approach for an IVP corresponding to mechanical oscillator with weak damping. Derive sequence of IVPs.	Example sheet 6: Q1 and Q2.
06/03/06	Finish off perturbation solution for mechanical oscillator with weak damping. Compare against exact solution. Highlight features: Including more terms into the expansion increases accuracy at fixed time but ultimately all perturbation solutions diverge. Reason for divergence: Superficially: terms proportional to powers of t in the solutions; more deeply: errors in the ODE accumulate. Nonlinear example: Nonlinear pendulum. Key feature: Length of period increases at larger amplitudes. This is captured by the perturbation expansion.	Example sheet 6: All questions.

Assessment:

The course will be examined in a three hour exam in May/June. There will also be an in-class test which will account for 10% of the final mark. The test will be held in the Simon Building (Engineering) in lecture theatres Simon 1 (1.34) and Simon 3 (3.40) on

Thursday 16th March at 4pm.

Further details are available in pdf format. Rich Hewitt is likely to ask you to do some coursework, too.

Handouts:

Please note a few corrections for previous handouts (the files above have already been corrected).

Page last modified: March 21, 2006

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