• Mathematical Methods & Modelling
  • Preface
  • Acknowledgements
  • Course Outline
    • Course Overview
    • Staff Members
    • A Blended Approach to Learning
    • Learning Management System
    • Timetable
    • Communication
    • Consultations
    • Course Background and Purpose
    • Course Content, Goals, and Outcomes
      • Course Learning Outcomes
      • Chapter @ref(moddiff): An Introduction to Modelling and Differential Equations
      • Chapter @ref(odefirst): First-Order Differential Equations and Models
      • Chapter @ref(odesecond): Second-Order Differential Equations and Oscillatory Models
      • Chapter @ref(odesystem): Systems of Ordinary Differential Equations
    • Tentative Class Schedule
      • Block 3
      • Block 4
    • Assessment
    • Assessment Schedule
      • Calculation of the final mark
    • Textbooks and Recommended Reading
      • Mapping to Zill and Cullen
  • Prerequisites
  • 1 An Introduction to Modelling and Differential Equations
    • 1.1 Modelling
      • The Modelling Process
      • Mathematical Models
    • 1.2 Differential Equations
    • Summary
  • 2 First-Order Differential Equations and Models
    • 2.1 Differential Equations and Ordinary Differential Equations
    • 2.2 Notation
    • 2.3 Classification of an Ordinary Differential Equation
      • 2.3.1 Classification by order
      • 2.3.2 Classification by linearity
      • 2.3.3 Classification by homogeneity
      • 2.3.4 Classification by coefficients
      • Classification at a glance
      • Classification of Differential Equations: Examples
    • 2.4 Solutions of Ordinary Differential Equations
      • 2.4.1 Concept of a Solution
      • 2.4.2 Solution Curves
      • 2.4.3 Direction Fields: Seeing Solutions Without Solving
    • 2.5 Methods of Solution for Ordinary Differential Equations
      • Euler’s Number and Ordinary Differential Equations
      • 2.5.1 The Method of Direct Integration
      • 2.5.2 The Method of Separation of Variables
    • 2.6 Initial Value Problems
    • 2.7 First-Order Models: The Formal Modelling Process
      • Conceptual and Mathematical Formulations
    • 2.8 Models involving Growth
      • Explore: the population playground
    • 2.9 Models involving Decay
      • The physical setting
      • Model 1: A Single Decaying Substance
      • Model 2: A Decay Chain Ending in a Stable Substance
      • Model 3: A Two-Step Decay Chain
      • Model 4: A Chain with a Constant Supply
  • 3 Second-Order Differential Equations and Oscillatory Models
    • 3.1 The Method of Undetermined Coefficients
      • 3.1.1 Homogeneous Linear Equations with Constant Coefficients
      • Case 1: Distinct real roots, \(b^{2} - 4 a c > 0\)
      • Case 2: A repeated real root, \(b^{2} - 4 a c = 0\)
      • Case 3: Complex conjugate roots, \(b^{2} - 4 a c < 0\)
      • Summary: the three cases
      • Explore: the discriminant decides everything
      • Additional Examples
      • Exercise
      • Selected Answers
      • Did you get it?
    • 3.2 The Non-Homogeneous Equation: Undetermined Coefficients
      • Superposition, the second time
      • The idea: guess the form
      • Choosing the ansatz
      • Duplication, and why the rule works
      • Worked Examples
      • Duplication has a physical name: resonance
      • Exercise
      • Selected Answers
      • Did you get it?
  • Tutorials
    • Tutorial 1: Review of Algebra and Calculus
    • Tutorial 2: Classification, Solutions, and Direction Fields
    • Tutorial 3: Direct Integration, Separation of Variables, and Initial Value Problems
    • Tutorial 4: Growth Models
    • Tutorial 5: Decay Models
  • References
  • Published with bookdown

Mathematical Methods and Modelling I

Mathematical Methods and Modelling I

APPM1026A

Abdul Hamid Carrim

Semester 2, 2026
[Updated: 2026-09-25]