Differential Equations

From slope fields to the heat, wave and Laplace equations.

About this course

Fourteen lectures that build differential equations from the ground up. It starts with what a differential equation actually says — slope fields, initial-value problems, and the first-order methods — then moves to linear equations, resonance, the Laplace transform and systems with their phase portraits. Fourier series and transforms provide the bridge to partial differential equations, where separation of variables unlocks the heat, wave and Laplace equations in turn. The course closes on classification and numerics.

What you will be able to do

  • Solve first-order ODEs by separation, integrating factors and exactness, and read solutions off a slope field
  • Classify equilibria of autonomous equations and determine their stability
  • Solve second-order linear ODEs across all three cases, and handle forcing and resonance
  • Apply the Laplace transform and analyse systems of ODEs with phase portraits
  • Expand functions in Fourier series and use separation of variables to solve the heat, wave and Laplace equations

What this course covers

  • Calculus prerequisites: derivative, integral, exponential
  • What a differential equation is and its family of solutions
  • The slope field (direction field)
  • Initial value problems
  • Existence and uniqueness (Picard-Lindelöf)
  • Newton's law of cooling — end-to-end worked example
  • Course roadmap and the central unifying idea
  • Fourier Series: Functions as Infinite Tone Sums
  • Orthogonality and Projection of Basis Tones
  • Worked Signal: The Square Wave Coefficients
  • Gibbs Phenomenon: Permanent Overshoot at Jumps
  • Worked Signal: The Sawtooth Wave
  • Symmetry Shortcuts: Even and Odd Functions
  • Parseval's Theorem and Energy Conservation
  • Separation of Variables Ansatz
  • Breadth of Separation of Variables
  • Eigenvalue Problem from Boundary Conditions
  • Time Factors and Mode Decay Rates
  • Superposition and Fourier Fit to Initial Condition
  • Master Template for Linear PDEs
  • Worked Example: Uniform Initial Temperature
  • Prerequisites: exponentials, discriminant, Euler's formula
  • The Characteristic Equation
  • Distinct Real Roots, Superposition, and Solution-Space Dimension
  • Complex Conjugate Roots and Decaying Oscillations
  • Repeated Root and the Factor of t
  • The Mass-Spring-Damper and the Damping Ratio ζ
  • Worked Suspension Design to Critical Damping
  • From Fourier Series to the Fourier Transform
  • Real-World Applications of the Fourier Transform
  • Operational Rules: Linearity, Shift, and Derivative Rule
  • Why Complex Exponentials Are Used
  • Gaussian Transform and the Uncertainty Principle
  • Convolution Theorem
  • Diffusion and the Heat Equation via the Transform
  • Calculus prerequisites for first-order methods
  • Separable equations
  • Linear first-order equations and the integrating factor
  • Exact equations and potential reconstruction
  • Diagnosis: choosing the right solution method
  • Mixing tank: linear method applied end-to-end
  • Spine clause two: form as lock, structure as key
  • Structure of the forced solution: YH plus YP
  • Undetermined coefficients
  • Overlap fix: multiplying the guess by T
  • Variation of parameters
  • Resonance: undamped runaway growth
  • Three faces of resonance: beats, runaway, and the resonance curve
  • Resonance magnification and the quality factor Q
  • Derivation of the Heat Equation
  • Universality of the Heat Equation
  • Finite-Bar Solution via Separation of Variables
  • Smoothing and Irreversibility of Heat Flow
  • The Heat Kernel on an Infinite Bar
  • Boundary Conditions and Mode Families
  • Worked Example: Heat Kernel Peak Decay
  • Deriving the wave equation from Newton's law
  • D'Alembert's solution via characteristic coordinates
  • Standing waves and normal modes on a fixed string
  • Standing waves as superposition of two traveling waves
  • Finite speed, light cone, and wave–heat duality
  • Heat, wave, and Laplace equations as one family
  • Fully worked plucked-string problem
  • Autonomous equations and equilibria
  • The phase line
  • Stability and the derivative test
  • Semi-stability and the derivative-test blind spot
  • Logistic growth model
  • Harvesting, bifurcation, and the tipping point
  • Maximum sustainable yield — worked fishery
  • Laplace Transform Definition and Convergence
  • Working Table, Linearity, and the First Shifting Rule
  • Derivative Rule and Initial Conditions
  • Inverse Transform via Partial Fractions
  • Step Function, Impulse, and the Second Shifting Rule
  • Convolution Theorem and Transfer Functions
  • Poles, Stability, and the Full IVP Workflow
  • The universal form: x dot equals Ax
  • The phase plane and trajectories
  • Solving x̂ = Ax via eigenvalues and eigenvectors
  • Classification of equilibria by eigenvalue type
  • The trace-determinant plane
  • Linearisation and the predator-prey capstone
  • Central result: destiny written in eigenvalues
  • Laplace's Equation as Steady State
  • Relaxation as a Numerical Method
  • Dirichlet Problem on a Rectangle
  • Dirichlet Problem on a Disk (Poisson Formula)
  • Mean Value Property and Maximum Principle
  • Elliptic Character and Global Dependence
  • Capstone: Center Temperature and Trilogy Summary
  • Classification by the Discriminant
  • Physical Worlds of Each PDE Type
  • Well-Posedness and Data Requirements
  • Finite Differences and Explicit Schemes
  • Stability Limits and the CFL Condition
  • Superposition of Modes as the Course Spine
  • Capstone Worked Problem and Course Takeaways

Syllabus

  1. First-Order ODEs & Dynamics

    1. What a Differential Equation Is — Slope Fields & Initial-Value Problems
    2. First-Order Methods — Separable, Linear & Exact
    3. Autonomous Equations, Equilibria & Stability

  2. Linear ODEs & Systems

    1. Second-Order Linear ODEs — The Three Cases
    2. Forcing, Undetermined Coefficients & Resonance
    3. The Laplace Transform
    4. Systems of ODEs & Phase Portraits

  3. Fourier & the Bridge to PDEs

    1. Fourier Series
    2. The Fourier Transform

  4. Partial Differential Equations

    1. Separation of Variables
    2. The Heat Equation
    3. The Wave Equation
    4. Laplace's Equation & Boundary-Value Problems
    5. Classification, Numerics & Course Capstone