Differential Equations

Contents

1. Introduction and Classification
2. First-Order ODEs
3. Second-Order Linear ODEs
4. Systems of ODEs
5. Laplace Transforms
6. Series Solutions
7. Fourier Series
8. Introduction to Partial Differential Equations
9. Stability and Phase Plane Analysis
10. Common Pitfalls
11. Problem Set

Overview

University-level differential equations notes covering ODEs, Laplace transforms, and stability analysis.

Topics Covered

• First-Order ODEs: Separable, linear, exact, integrating factors
• Second-Order Linear ODEs: Homogeneous, non-homogeneous, characteristic equation
• Laplace Transforms: Properties, inverse transforms, solving ODEs
• Stability Analysis: Phase planes, equilibrium points, linearisation

Prerequisites

• Single-variable calculus (differentiation, integration)
• Linear algebra (eigenvalues, matrices)
• Mathematical proofs and logic

How to Use These Notes

Start with first-order ODEs to build foundational knowledge, then progress to second-order and systems. Each section includes worked examples and practice problems.

Navigation

Use the sidebar to browse topics, or start with the introductory pages linked from the sidebar.

Additional Resources

Each section includes:

• Detailed explanations of key concepts
• Worked examples with step-by-step solutions
• Practice problems with answers
• Common pitfalls and how to avoid them
• Connections to other areas of mathematics

Study Tips

1. Master the methods: Learn to identify and solve different types of ODEs
2. Practise regularly: Differential equations require active practice
3. Draw phase portraits: Visualise solutions for systems of ODEs
4. Learn standard examples: Know the properties of common equations (harmonic oscillator, exponential decay)
5. Connect to applications: Relate differential equations to physics, biology, and engineering