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Differential Equations | Mathematics

sources:

  • text: Spivak - Calculus

Differential Equations

Section titled “Differential Equations”

Contents

Section titled “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

Section titled “Overview”

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

Topics Covered

Section titled “Topics Covered”
  • First-Order ODEs: Separable, linear, exact, integrating factors. These are the building blocks — mastering first-order techniques is essential for higher-order equations.
  • Second-Order Linear ODEs: Homogeneous, non-homogeneous, characteristic equation. These model oscillations (springs, circuits) and appear throughout physics and engineering.
  • Laplace Transforms: Properties, inverse transforms, solving ODEs. The Laplace transform converts differential equations into algebraic equations, solving initial value problems systematically.
  • Stability Analysis: Phase planes, equilibrium points, linearisation. Stability analysis asks whether equilibria attract or repel nearby trajectories, revealing qualitative behaviour without solving explicitly.

Prerequisites

Section titled “Prerequisites”
  • Single-variable calculus (differentiation, integration). You must be comfortable with techniques of integration, including substitution and partial fractions.
  • Linear algebra (eigenvalues, matrices). Systems of ODEs require eigenvalue methods; matrix exponentials solve linear systems.
  • Mathematical proofs and logic. Existence and uniqueness theorems require rigorous argumentation.

How to Use These Notes

Section titled “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. The key is to learn to classify ODEs and identify the appropriate solution method.

Section titled “Navigation”

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

Additional Resources

Section titled “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

Intuition

Section titled “Intuition”

Differential equations describe how systems evolve, from planetary orbits to population dynamics. The core insight is that knowing the current state and the laws of motion determines the future: an ODE encodes rates of change as relationships between a function and its derivatives. First-order equations model exponential growth and decay, while second-order equations capture oscillations and resonance. Laplace transforms convert differential equations into algebraic ones, trading calculus for algebra. Stability analysis asks whether equilibria attract or repel nearby trajectories, visualised through phase portraits that reveal the qualitative behaviour of solutions without solving explicitly.

Study Tips

Section titled “Study Tips”
  1. Master the methods: Learn to identify and solve different types of ODEs. Classification is the first step in any solution.
  2. Practise regularly: Differential equations require active practice. You cannot learn to solve ODEs by reading alone.
  3. Draw phase portraits: Visualise solutions for systems of ODEs. Phase plane analysis reveals qualitative behaviour.
  4. Learn standard examples: Know the properties of common equations (harmonic oscillator, exponential decay). These are the building blocks.
  5. Connect to applications: Relate differential equations to physics, biology, and engineering. Applications provide motivation and context.

Cross-References

Section titled “Cross-References”