Differential Equations

MATH2352 Interactive Lecture Notes

AI-generated interactive notes covering first and second order ODEs. Each page includes theory, worked examples, and practice questions with step-by-step solutions.

Week 1 — First Order Equations: Introduction

1.1 What are Differential Equations? Exercises

Classification, order, linearity, direction fields, initial value problems

1.2 Partial Derivatives Supplementary

Review of multivariable calculus needed for exact equations

1.3 Modelling with First Order Equations Exercises

Population growth, mixing problems, Newton's cooling law, falling bodies

1.4 Integrating Factors Exercises

Linear first order equations, integrating factor method, applications

Week 2 — First Order Equations: Methods

2.1 Separation of Variables Exercises

Separable equations, substitution techniques, Bernoulli equations

2.2 Euler's Numerical Method Exercises

Forward Euler, step size, error analysis, improved Euler method

2.3 Exact and Non-exact Equations Exercises

Exactness condition, solving exact ODEs, integrating factors for non-exact

Week 3 — Second Order Linear Equations

3.1 Homogeneous Equations Exercises

Characteristic equation, real/complex/repeated roots, superposition principle

3.2 Wronskians and Linear Independence Exercises

Wronskian determinant, fundamental sets, Abel's theorem

Week 4 — Second Order: Non-homogeneous

4.1 Repeated Characteristic Roots Exercises

Reduction of order, repeated roots, Euler-Cauchy equations

4.2 Non-homogeneous Equations Exercises

Method of undetermined coefficients, particular solutions

4.3 Variation of Parameters Exercises

General method for non-homogeneous equations, applications