*Section 102: September-December 2014*

Boyce and DiPrima, which is the book for this course, can be found online here. This is the 9th edition, but it is almost identical to the 10th edition.

** Mid-term examination 1:** Worked answers here.

** Mid-term examination 2:** Worked answers here.

### General (non-homework) Links

Brief summary notes on linear 2nd-order ODEs

Brief notes with worked examples on Laplace Transforms

Notes on Separation of Variables, with examples for Wave/Diffusion/Laplace equation.

### Homework

Homework 1: Set on Monday 8th September. Hand in 10am Monday 15th September. Suggestions for additional practice questions can be found at the end of sheet. Answers here.

Homework 2: Set on Monday 15th September. Hand in 10am Monday 22nd September. Suggestions for additional practice questions can be found at the end of sheet. Answers here.

Homework 3: Set on Monday 22nd September. Hand in 10am Monday 29th September. Suggestions for additional practice questions can be found at the end of sheet. Answers here.

Example questions 4: Set on Monday 29th September. Do not hand in - these are just practice questions. Worked answers can be found here.

Example questions 5: Set on Monday 6th October. Do not hand in - these are just practice questions. Worked answers can be found here.

Homework 6: Set on Monday 13th October. Hand in 10am Monday 20th October. Suggestions for additional practice questions can be found at the end of the sheet. Answers here.

Homework 7: Set on Monday 20th October. Hand in 10am Monday 27th October. Suggestions for additional practice questions can be found at the end of the sheet. Answers here.

Homework 8: Set on Monday 27th October. Hand in 10am Monday 3rd November. Suggestions for additional practice questions can be found at the end of the sheet. Answers here.

Homework 9: Set on Monday 3rd November. Hand in 10am Monday 10th November. Suggestions for additional practice questions can be found at the end of the sheet. Answers here.

Example questions 10: Set on Monday 10th November. Do not hand in - these are just practice questions. Worked answers can be found here.

Homework 11: Set on Monday 17th November. Hand in 10am Wednesday 26th November. Suggestions for additional practice questions can be found at the end of the sheet. Worked answers can be found here.

Homework 12: Set Wednesday 26th November. ONLY 1 HOMEWORK QUESTION (due Monday 1st December, between 09:00 and 14:00 outside LSK203, or hand in on Friday 28th November in class). The rest of the questions are more challenging practice questions, with answers here.

FINAL EXAM PRACTICE QUESTIONS: Answers here.

### Broad-brush lecture outline, week by week

* Relevant section numbers in Boyce and DiPrima given in square brackets []*

Week 1 (to Friday 5th September): **Main concept: Introduction to differential equations.** Terminology of differential equations [1.3]

Week 2 (to Friday 12th September): ** Main concept: First-order ODEs, and how to solve them.** How to solve separable equations [2.2] and using the method of integrating factors [2.1]. Being able to convert from words to math [2.3]. Difference between linear/non-linear equations [2.4]. Autonomous equations: how to learn about the nature of the solutions without having to solve the equation [2.5].

Week 3 (to Friday 19th September): ** Main concept: (Linear) Second-order ODEs, and how to solve them. ** Method for solving homogeneous linear second-order equations with constant coefficients [3.1]. Some general theory for homogeneous linear second-order equations - the Wronskian, and the concept of 'fundamental sets of solutions' [3.2]. Solving equations with either complex roots [3.3], or repeated roots [3.4].

Week 4 (to Friday 26th September): ** Main concept: (Linear) Second-order ODEs: Solutions for non-homogeneous equations. ** Finish material on repeated roots of equations, and general theory for finding a second root by 'reduction of order' [3.4]. Non-homogeneous equations: Method of undetermined coefficients [3.5]; and Method of variation of parameters [3.6]. Mathematical modelling: applications of 2nd-order DEs [3.7-3.8] (briefly).

Week 5 (to Friday 3rd October): ** Main concept: Finish off 2nd-order ODEs, (maybe) start thinking about Systems of Equations, (definitely) sit 1st mid-term exam. ** Finish Method of variation of parameters [3.6], and briefly mention Mathematical Modelling [3.7-3.8]. Summarize before exam. (maybe) Start looking at systems of linear equations [7.1]

Week 6 (to Friday 10th October): ** Main concept: Systems of linear 1st-order equations: basic ideas, homogeneous problems with distinct real roots. ** Review of basic linear algebra: eigenvectors and eigenvalues [7.2-7.4], homogeneous systems of 1st-order equations: distinct real roots and the concept of phase portraits [7.5], complex roots [7.6].

Week 7 (to Friday 17th October): ** Main concept: Systems of linear 1st-order equations: homogeneous systems with repeated roots. ** Finish complex roots [7.6], (briefly) definition of fundamental matrix [7.7], start repeated roots [7.8]. Note: there is no class on either Monday 13th or Wednesday 15th.

Week 8 (to Friday 24th October): ** Main concept: Systems of linear 1st-order equations: repeated roots, and non-homogeneous systems. ** Repeated roots [7.8], non-homogeneous equations [7.9] - three methods for solving these: undetermined coefficients; diagonalisation; or variation of parameters.

Week 9 (to Friday 31st October): ** Main concept: Laplace transforms. ** Basic definition and some simple Laplace transforms [6.1]; application to ODEs [6.2]; use with step functions and application to ODEs with discontinuous forcing [6.3,6.4].

Week 10 (to Friday 7th November): ** Main concept: More Laplace transforms, and introduction to PDEs. ** Impulse functions (Dirac delta function) [6.5] and convolution [6.6]. Begin partial differential equations [chapter 7] - basic ideas [7.1].

Week 11 (to Friday 14th November): ** Main concept: Fourier Series. ** Boundary value problems, eigenfunctions/eigenvalues [7.1]; Fourier series: periodicity, odd/even functions, orthogonality, method for calculating Fourier series [7.2 - 7.4].

Week 12 (to Friday 21st November): ** Main concept: MID-TERM 2. Separation of variables for PDEs. ** Finish Fourier series [10.3,10.4]. Separation of variables: diffusion equation (heat conduction) [10.5,10.6].

Week 13 (to Friday 28th November): ** Main concept: Separation of variables for PDEs ** Wave equation (vibrations of a string) [10.7], Laplace's equation [10.8]. END OF COURSE.

Contact me: drh39 'at' cam.ac.uk