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      Announcements Forum
      GENERAL ANNOUNCEMENTS

    • Forum icon
      Student forum

      STUDENT FORUM


      This forum is available for everyone to post messages to.

      If you have any questions about the module, please ask them here.

      Students are encouraged to post questions to this forum more than writing emails directly to the Professor.

      The forum will be monitored Tuesday and Thursday after 5pm.

      Students should feel free to reply to other students if they are able to. Group learning and peer discussion are very helpful for active learning of new knowledge.

    • Quiz icon
      Associate Students Exam 2023-2024 Quiz

      This handwritten assessment is available for a period of 3 hours, within which you must submit your solutions. You may log out and in again during that time,but the countdown timer will not stop. If your attempt is still in progress at the end of your 3 hours, any file you have uploaded will be automatically submitted.

      In completing this assessment:

      •You may use books and notes.

      •You may use calculators and computers, but you must show your working for any calculations you do.

      •You may use the Internet as a resource, but not to ask for the solution to an exam question or to copy any solution you find.

      •You must not seek or obtain help from anyone else.


      Not available unless: You belong to Associates 2023-2024
    • File icon
      MATHEMATICA EXAMPLE OF ODE SOLVER File
      338.0 KB

      HELPFUL MATHEMATICA SOFTWARE

      Simple examples of ODEs solver with "Mathematica"software.

      Mathematica, MATLAB, etc. are useful softwares to check the analytical solution of differential equations. There are free Mathematica licenses for students in QMUL, which you can require by contact QMUL IT following the QMUL Mathematica software webpage.

      https://www.its.qmul.ac.uk/support/self-help/software/free-and-discounted-software/mathematica/




  • Week 1 - Introduction, Separable 1st-order ODEs

     WEEK  1 PREPARATORY TASKS


    WEEK 1 SUBJECT

    Introduction of Ordinary Differential Equation. Separable 1st-order ODEs. Reducible to separable 1st-order ODE (z=ax+by+c)


    WEEK 1 ACTIVITIES


    WEEK 1 LECTURES

    To follow online the live lectures and tutorials login to the Online course room


    LINKS TO HANDWRITTEN NOTES AND SLIDES (will be avaible after the live lessons)

  • Week 2 - First order ODEs


    WEEK 2 SUBJECT

    Scale invariant  1st-order ODE (reducible to separable), homogenous 1st-order Linear ODE, inhomogenous 1st-order Linear ODE (Variation of parameter method), Exact 1st-order ODE

    Most Exact 1st-order ODEs are non-linear as well, and they are in general not exact. In this module,  we consider exclusively separable or exact ODEs.

    WEEK 2 ACTIVITIES


    WEEK 2 LECTURES AND TUTORIALS

    To follow online the live lectures and tutorials login to the Online course room


    LINKS TO HANDWRITTEN NOTES AND SLIDES (will be available after the live lessons)

  • Week 3 - Initial Value Problem (I.V.P) and Picard-Lindelöf Theorem of 1st-order ODEs


    WEEK 3 SUBJECT

    Initial Value Problem (I.V.P), Picard-Lindelöf Theorem (existence and uniqueness of the solutions of I.V.Ps of the 1st-order ODE).
    Transformation of  a nth-order ODE to a system of  1-st order  ODEs.

    WEEK 3 ACTIVITIES


    WEEK 3 LECTURES AND TUTORIALS

    To follow online the live lectures and tutorials login to the Online course room

    • Lesson 1: Initial Value Problem (I.V.P.) and motivational examples for the Picard-Lindelöf Theorem
    • Lesson 2: Picard-Lindelöf Theorem
    • Lesson 3: Transformation of  a nth-order ODE to a system of 1-st order  ODEs.
    • Tutorial: Covering   Formative Assessment Week 3  (Solutions) and Mock Quiz Week 3

    LINKS TO HANDWRITTEN NOTES AND SLIDES (will be avaible after the live lessons)


  • Week 4 - Homogeneous 2nd-order ODEs and I.V.P of 2nd-order ODEs


    WEEK 4 SUBJECT

    Obtaining the general solutions to homogeneous 2nd-order linear ODEs (by characteristic equations), and solve I.V.P to 2nd-order linear ODEs.

    WEEK 4 ACTIVITIES


    WEEK 4 LECTURES AND TUTORIALS

    To follow online the live lectures and tutorials login to the Online course room

    • Lesson 1: General introduction to 2nd order linear ODE
    • Lesson 2: Solution of 2nd order linear ODEs with constant coefficients
    • Lesson 3: Solution of 2nd order linear ODEs with constant coefficients and IVP
    • Tutorial : Covering   Formative Assessment Week 4  (Solutions) and Mock Quiz Week 4


    LINKS TO HANDWRITTEN NOTES AND SLIDES (will be avaible after the live lessons)


  • Week 5 - Euler type equations, inhomogenerous 2nd-order ODEs, Educated Guess Method

    WEEK 5 SUBJECT

    Euler type equations, variation of parameter method  for inhomogenerous 2nd-order ODEs, educated guess method

    WEEK 5 ACTIVITIES


    WEEK 5 LECTURES AND TUTORIALS

    To follow online the live lectures and tutorials login to the Online course room


    LINKS TO HANDWRITTEN NOTES AND SLIDES (will be avaible after the live lessons)


  • Week 6 - Boundary Value problem (B.V.P.) of 2nd-order ODEs , the Theorem of The Alternative

    WEEK 6 SUBJECT

    Introduction to B.V.P. ,  Theorem of the Alternative (theorem of the existence and uniqueness of solutions of B.V.Ps)

    WEEK 6 ACTIVITIES

    • Participate in the live lectures
    • Read Week 6 Lecture notes 
    • Answer the   Formative Assessment Week 6 Practice and exploration questions
    • Train with Mock Quiz Week 6

    • Read about a notable mathematician in differential equations  the Argentinian-American Mathematician (AbelPrize 2023) Luis Caffarelli (Wikipedia page)  whose work includes equations underpinning physical phenomena, such as melting ice and flowing liquids as outlined in this excellent Nature article. Check out his interesting profile on MacTutor!

    • Complete Coursework 1  (Quiz) on the material of week 1-6. The Coursework will be open starting from Friday 3 November 2023 12pm Deadline: Friday 10 November 2023 12pm.  Once opened you have 48 hours to complete the work. This is a summative  quiz based assessment that counts 10% towards your module mark.  All late submissions will be given 0 marks if the student does not have approved EC. You have only one attempt at the assessment. If your attempt at the coursework is still in progress at the end of the allowed time, the answers you have filled in so far will be automatically submitted. You should read the Important information about coursework and tutorials before attempting this assessment.

    WEEK 6 LECTURES AND TUTORIALS

    To follow online the live lectures and tutorials login to the Online course room

    • Lesson 1:Introduction to Bounday Value Problems (B.V.P.)
    • Lesson 2: Introduction to the Theorem of the Alternative
    • Lesson 3:  Theorem of the Alternative and its applications.
    • Tutorial : Covering   Formative Assessment Week 6 Practice and exploration questions (Solutions) and Mock Quiz Week 6


    LINKS TO HANDWRITTEN NOTES AND SLIDES (will be avaible after the live lessons)


    • Quiz icon
      Coursework 1 Quiz

       Coursework 1  (Quiz) on the material of week 1-6. The Coursework will be open starting from Friday 3 November 2023 12pm Deadline: Friday 10 November 2023 12pm.  Once opened you have 48 hours to complete the work. This is a summative assessment (in the form of a QMPLUS quiz) that counts 10% towards your module mark.  You have only one attempt at the assessment. All late submissions will be given 0 marks if the student does not have approved EC. If your attempt at the quiz is still in progress at the end of the allowed time, the answers you have filled in so far will be automatically submitted. You should read the Important information about coursework and tutorials before attempting this assessment.

      Solutions will be available after the deadline.


    • File icon
      Solution to Coursework 1 File
      127.9 KB
    • File icon
      Solution to question 3 Coursework 1 File
      586.3 KB

  • Week 7 - Past papers

  • Week 8 - Autonomous Systems, Dynamical Systems, Equilibria, Linearisation of systems of nonlinear ODEs

    WEEK 8 SUBJECT

    Autonomous systems, Dynamical systems, Equilibria, Linearisation of systems of nonlinear ODEs.

    WEEK 8 ACTIVITIES

     
    WEEK 8 LECTURES AND TUTORIALS

    To follow online the live lectures and tutorials login to the Online course room

    • Lesson 1:Autonomous Systems, Dynamical Systems, IVP of dynamical systems.
    • Lesson 2: Trajectories, Equilibria
    • Lesson 3:  Linearization of a non-linear system of ODEs
    • Tutorial 1 &2: Covering   Formative Assessment Week 8 Practice and exploration questions (Solutions)  Mock Quiz Week 8

     

    LINKS TO  HANDWRITTEN NOTES AND SLIDES (will be avaible after the live lessons)

  • Week 9 - Solving linear ODE systems, Eigenvalues and Eigenvectors, Introduction to Phase portraits

    WEEK 9 SUBJECT

    Solving linear ODE systems, Eigenvalues and Eigenvectors, Introduction to  Phase Portraits

    WEEK 9 ACTIVITIES

    WEEK 9 LECTURES AND TUTORIALS

    To follow online the live lectures and tutorials login to the Online course room

    • Lesson 1: Revision of Algebra, Eigenvalues and Eigenvectors
    • Lesson 2: Solving a linear system of first order ODEs
    • Lesson 3:  Phase portraits:introduction and take home message
    • Tutorial 1 &2: Covering   Formative Assessment Week 9 Practice and exploration questions (Solutions) and Mock Quiz Week 9


    LINKS TO  HANDWRITTEN NOTES AND SLIDES (will be avaible after the live lessons)


  • Week 10- Phase portraits of linearised systems

    WEEK 10 SUBJECT

    Phase portrait of linearised systems. Case of real eigenvalues of the linearised system. Case of complex eigenvalues of the linearised system.

    WEEK 10 ACTIVITIES


    WEEK 10 LECTURES AND TUTORIALS

    To follow online the live lectures and tutorials login to the Online course room

    • Lesson 1:Phase portraits: case of real and distinct eigenvalues (saddle, stable and unstable node)
    • Lesson 2: More on phase portraits
    • Lesson 3: Phase portraits: case of  complex eigenvalues (stable and unstable focus, centre
    • Tutorial 1 &2: Covering  Formative Assessment Week 10 Practice and exploration questions (Solutions) Mock Quiz Week 10


    LINKS TO HANDWRITTEN NOTES AND SLIDES (will be avaible after the live lessons)

  • Week 11-Lyapunov and asymptotic stability criteria

    WEEK 11 SUBJECT

    Summary of phase portraits for linearised systems. Lyapunov and  asymptotic stability. Lyapunov function.

    WEEK 11 ACTIVITIES

    • Participate in the live lectures
    • Read Week 11 Lecture notes
    • Answer the  Formative Assessment Week 11
    • Train with Mock Quiz Week 11
    • Read about a notable mathematician in partial differential equations differential geometry and mathematical physics: the Abel Prize Prof. Karen Uhlenbeck (Wikipedia page) first woman to get this important award. Check out her interesting profile on MacTutor !
    • Coursework 2  (Quiz) on the material of week 8-11. The Coursework will be open starting from Friday 8 December 2023 1pm Deadline: Friday 15 December 2023 1pm.  Once opened you have 48 hours to complete the work. This is a summative assessment (in the form of a QMPLUS quiz) that counts 10% towards your module mark.  All late submissions will be given 0 marks if the student does not have approved EC. You have only one attempt at the assessment. If your attempt at the quiz is still in progress at the end of the allowed time, the answers you have filled in so far will be automatically submitted. You should read the Important information about coursework and tutorials before attempting this assessment.
    WEEK 11 LECTURES AND TUTORIALS

    To follow online the live lectures and tutorials login to the Online course room


    LINKS TO HANDWRITTEN NOTES AND SLIDES (will be avaible after the live lessons)


    • Quiz icon
      Coursework 2 Quiz

      Coursework 2 (Quiz) on the material of week 8-11. The Coursework will be open starting from Friday 8 December 2023 1pm Deadline: Friday 15 December 2023 1pm.  Once opened you have 48 hours to complete the work.This is a summative assessment (in the form of a QMPLUS quiz) that counts 10% towards your module mark.  You have only one attempt at the assessment. All late submissions will be given 0 marks if the student does not have approved EC. If your attempt at the quiz is still in progress at the end of the allowed time, the answers you have filled in so far will be automatically submitted. You should read the Important information about coursework and tutorials before attempting this assessment.

      Solutions will be available after the deadline.


    • File icon
      Solutions to Coursework 2 File
      121.4 KB

  • Week 12 - Rest of teaching content and revision week

    WEEK 12 SUBJECT

    Revision of the course content. Lesson 3 and  Tutorials  are revision and question time,  students can use the forum to ask questions and  revision of some material they would like to see covered or they can either  ask questions directly to the Professor during the interactive sessions.

    WEEK 12 ACTIVITIES

    • Participate in the live lectures 
    • Make an effort to go  through all lecture contents and try to identify and discuss your learning difficulties with the Professor in interactive sessions.
    • Complete Coursework 2 (Assessed Quiz ) on the material of week 8-11. Please submit by  Friday 15 December 2022 at 1pm. This is a summative assessment (in the form of a QMPLUS quiz) that counts 10% towards your module mark.  You have only one attempt at the assessment. If your attempt at the quiz is still in progress at the end of the allowed time, the answers you have filled in so far will be automatically submitted. You should read the Important information about coursework and tutorials before attempting this assessment.


    WEEK 12 LECTURES AND TUTORIALS

    To follow online the live lectures and tutorials login to the Online course room

    • Lesson 1: Revision, Exam preparation
    • Lesson 2:  Revision, Exam preparation
    • Lesson 3:  Revision,Exam preparation
    • Tutorial 1 : Revision,Exam preparation, Question time


    LINKS TO RECORDINGS, HANDWRITTEN NOTES AND SLIDES (will be avaible after the live lessons)



  • Assessment information

    • Assessment Pattern-

      2 Courseworks (20% of final mark)  + Final Exam (80% final mark)

      Format and dates for the in-term assessments -

      Coursework 1 (10% final mark) 
      Quiz based coursework on material of week 1-6
      Opens: Friday 3 November 2023 12pm Deadline: Friday 10 November 2023 12pm

      Coursework 2 (10% final mark) 
      Quiz based coursework on material of week 8-11
      Opens: Friday 8 December 2023 12pm Deadline Friday 15 December 2023

      Format of final assessment -

      Final Exam (On Campus 80% final mark)
      Questions 1-3: Material of weeks 1-6
      Question 4: Material of weeks 8-12

      link to past papers -

      Past papers are available  under Module Content Week 7

      Description of Feedback -

      Feedback on the Coursework (Quizzes) will be available soon after the coursework deadline.
      Feedback on Formative Assessment will be provided during Tutorials and Support Learning Hours and through the Module Forum

       

  • Q-Review

  • Module description

  • Syllabus

    MTH5123 Differential Equations

     Introduction to ordinary differential equations (ODE)

    • Dependent and independent variables, 
    • Normal forms for differential equations
    • Order of differential equations
    • Homogeneous and Inhomogeneous ODEs

     First order differential equations

    • Separable ODE
    • Reducible to separable ODE
    • Linear 1st-order differential equations and variation of parameter method
    • Exact 1st-order differential equations
    • Initial Value Problem and the Picard-Lindelof  theorem

     Secodn-order differential equations

    • Linear second order differential equations
    • Euler differential equations
    • Inhomogeneous 2nd order ODE: Educated guess method
    • Reduction of higher-order differential equation to a first order system of ODEs
    • Boundary Value Problem and The theorem of the Alternative

     Systems of ordinary differential equations

    • Autonomous systems, Dynamic systems
    • Stationary solutions, Fixed points, Linearisation of system of ODEs
    • Solutions of linear systems of differential equations, Eigenvalues, Eigenvectors
    • Linear stability analysis
    • Phase potraints
    • Lyapunov functions and stability of non-linear systems of differential equations

  • Module aims and learning outcomes

    • Module Description

      Differential equations frequently arise in applications of mathematics to science, engineering, social science, biology, medicine and economics. This module provides an introduction to the methods of analysis and solution of simple classes of ordinary differential equations. The topics covered will include first- and second-order differential equations, autonomous systems of differential equations and analysis of stability of their solutions.


      Learning Aims and Outcomes

      ACADEMIC CONTENT

      This module covers:

      • Simple types of first-order and second-order ordinary differential equations (ODEs).
      • Autonomous systems of first-order ODEs, finding their solutions close to equilibrium points and plot phase portraits (trajectories) of an ODE system.

      DISCIPLINARY SKILLS

      At the end of this module, students should be able to:

      • Identify and solve first-order ordinary differential equations (ODEs) that are separable, exact or linear, or can be reduced to the above by standard methods.
      • Identify and solve linear second-order non-homogeneous differential equations with constant coefficients and associated initial value problems.
      • Identify and investigate boundary value problems for linear second-order ODEs.
      • Identify and solve general systems of first-order linear differential equations with constant coefficients using matrix operations.
      • Explain the notions of phase space, trajectories, and equilibria for autonomous systems of two first-order differential equations.
      • Explain the notions of stability for general systems of differential equations and apply a given Lyapunov function for investigating the stability of simple nonlinear systems of differential equations.
      • Find equilibria of a given autonomous system of differential equations and describe the system's behaviour and sketch the phase portrait close to an equilibrium in linear approximation.

      ATTRIBUTES

      At the end of this module, students should have developed with respect to the following attributes:

      • Acquire and apply knowledge in a rigorous way.
      • Connect information and ideas within their field of study.
      • Grasp the principles and practices of their field of study.
      • Apply their analytical skills to investigate unfamiliar problems.
      • Acquire substantial bodies of new knowledge.

    • MODULE EXPECTATIONS

      TEACHING ARRANGEMENTS

      Each lecture is based on the week’s topic.

      Formative assessments and mock quizzes

      There are in total 10 Formative Assessements and 10 Mock Quizzes for this module. Each Formative Assessement and each Mock Quiz refers to one week learning content of this module. They are very  important for you to prepare for the assessed courseworks and for the final exam and will allow you to  practice methods, find and solve learning difficulties, ask questions in tutorials or interactive online lectures.

      How you engage with Formative Assessments, is therefore crucial to your success.


      Courseworks

      There are in total 2 courseworks for this module. The first  coursework refers to the material of weeks 1-3 the second coursework refers to the material of weeks 4-6. Each coursework counts as 10% of the total mark of this module.

      How you engage with courseworks, is therefore crucial to your success.

      PREPARING FOR THIS MODULE

      • Read the assigned reading thoroughly. Make good notes – don’t just highlight.
      • Review lecture notes and use one of the recommended survey books to enhance your understanding of the week’s topic. 
      • Practice the Formative Assessements independently and BRING questions  to the  tutorial sessions.

      Tip: The learning content will become more difficult through the learning weeks. Your goal should be

      1. To fully understand the learning content at the end of each week.

      2. Practice all coursework questions before the lectures of next week and

      3. Master all learning difficulties of the past week after discussion with the lecture and other tutors.


  • Assessment

  • Formative assessments

  • Teaching team

  • Hints and tips

  • Where to get help

    There may be times during the term when you get stuck doing your homework or project. This is normal. 

    Who to contact for what:

    Should you find learning difficulties, bring those questions to the tutorials or to the Support Learning Hours

    Alternatively you can  post your question to the online forum.

    The forum is monitored twice a week by the Module Lead, moreover you can discuss with the other students of the module about the module content.

    Try to limit direct emails to the Module Lead at the minimum to allow the Module Lead to answer all emails in time

    Module Lead: g.bianconi@qmul.ac.uk



  • Module handbook

  • Lecture notes

  • Coursework

  • Exam papers

  • General information on the Final Exam

    • File icon
      Appendix File
      83.2 KB

    • IMPORTANT INFORMATION ABOUT THE FINAL EXAM

      The final exam of MTH5123 will take place in in person and it will be handwritten.

      It will comprise 4 sections (Questions) each one requiring handwritten answer to several points.

      Exam content:

      Question 1-3 will focus on the material of weeks 1-6

      Question 4 will focus on the material of week 8-11

      You will be tested in your ability to solve ODEs and in your problem solving skills.

      You will not be asked to prove any derivation but questions related to the Picard Lindelof theorem and the Theorem of the Alternative might be present.

      You will not be asked to sketch the solutions of ODEs but you might be asked to sketch phase potraits of systems of ODEs.

      A good training for the final exam are past papers (list in the week 7 tab), expecially the 2018 exam paper.

  • Mock quizzes

  • Generated by Assessment Information block

     In 2023/2024, the assessment structure will be 2 online courseworks (20% of the total mark ) and a in person final exam (80% of the total mark). For your final assessment, the marking criteria gives credit both for (clearly explained) method and final answer. You will learn what are considered clearly explained methods through the lectures during the whole semester. You can ask questions during tutorial and you will get personalized feedback during the tutorial and the Support Learning Hours. The forum is also a valid resource to communicate your questions to your fellow colleagues. The forum will be monitored by the Professor twice a week providing feedback and answers to the questions raised.

    Assessment Structure:
    Coursework 1 (10% final mark) 
    Quiz based coursework on material of week 1-6
    Opens: Friday 3 November 2023 12pm Deadline: Friday 10 November 2023 12pm

    Coursework 2 (10% final mark) 
    Quiz based coursework on material of week 8-11
    Opens: Friday 8 December 2023 1pm Deadline Friday 15 December 2023 1pm

    Final Exam (On Campus 80% final mark)
    Questions 1-3: Material of weeks 1-6
    Question 4: Material of weeks 8-12

    Feedback on the Coursework will be available soon after the coursework deadline.

    Feedback on Formative Assessment will be provided during Tutorials and Support Learning Hours and through the Module Forum

  • Reading List Online